| L(s) = 1 | + 10·2-s + 55·4-s + 4·7-s + 220·8-s − 2·11-s + 6·13-s + 40·14-s + 715·16-s − 20·22-s + 21·25-s + 60·26-s + 220·28-s − 16·31-s + 2.00e3·32-s + 4·41-s − 110·44-s − 22·49-s + 210·50-s + 330·52-s − 22·53-s + 880·56-s + 58·61-s − 160·62-s + 5.00e3·64-s − 8·77-s + 40·82-s − 22·83-s + ⋯ |
| L(s) = 1 | + 7.07·2-s + 55/2·4-s + 1.51·7-s + 77.7·8-s − 0.603·11-s + 1.66·13-s + 10.6·14-s + 178.·16-s − 4.26·22-s + 21/5·25-s + 11.7·26-s + 41.5·28-s − 2.87·31-s + 353.·32-s + 0.624·41-s − 16.5·44-s − 3.14·49-s + 29.6·50-s + 45.7·52-s − 3.02·53-s + 117.·56-s + 7.42·61-s − 20.3·62-s + 625.·64-s − 0.911·77-s + 4.41·82-s − 2.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 113^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{20} \cdot 113^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10674.24138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10674.24138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{10} \) |
| 3 | \( 1 \) |
| 113 | \( 1 - 20 T + 361 T^{2} - 4560 T^{3} + 57958 T^{4} - 564856 T^{5} + 57958 p T^{6} - 4560 p^{2} T^{7} + 361 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \) |
| good | 5 | \( 1 - 21 T^{2} + 229 T^{4} - 1816 T^{6} + 11858 T^{8} - 64966 T^{10} + 11858 p^{2} T^{12} - 1816 p^{4} T^{14} + 229 p^{6} T^{16} - 21 p^{8} T^{18} + p^{10} T^{20} \) |
| 7 | \( ( 1 - 2 T + 17 T^{2} - 31 T^{3} + 136 T^{4} - 270 T^{5} + 136 p T^{6} - 31 p^{2} T^{7} + 17 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 11 | \( ( 1 + T + 20 T^{2} + 54 T^{3} + 3 p^{2} T^{4} + 554 T^{5} + 3 p^{3} T^{6} + 54 p^{2} T^{7} + 20 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 13 | \( ( 1 - 3 T + 15 T^{2} + 28 T^{3} + 132 T^{4} + 190 T^{5} + 132 p T^{6} + 28 p^{2} T^{7} + 15 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 17 | \( 1 - 82 T^{2} + 3629 T^{4} - 113023 T^{6} + 2690670 T^{8} - 50902486 T^{10} + 2690670 p^{2} T^{12} - 113023 p^{4} T^{14} + 3629 p^{6} T^{16} - 82 p^{8} T^{18} + p^{10} T^{20} \) |
| 19 | \( 1 - 103 T^{2} + 5774 T^{4} - 218582 T^{6} + 6139841 T^{8} - 132315942 T^{10} + 6139841 p^{2} T^{12} - 218582 p^{4} T^{14} + 5774 p^{6} T^{16} - 103 p^{8} T^{18} + p^{10} T^{20} \) |
| 23 | \( 1 - 134 T^{2} + 8017 T^{4} - 284551 T^{6} + 7128794 T^{8} - 159818442 T^{10} + 7128794 p^{2} T^{12} - 284551 p^{4} T^{14} + 8017 p^{6} T^{16} - 134 p^{8} T^{18} + p^{10} T^{20} \) |
| 29 | \( 1 - 154 T^{2} + 13145 T^{4} - 754288 T^{6} + 32237070 T^{8} - 1057564204 T^{10} + 32237070 p^{2} T^{12} - 754288 p^{4} T^{14} + 13145 p^{6} T^{16} - 154 p^{8} T^{18} + p^{10} T^{20} \) |
| 31 | \( ( 1 + 8 T + 85 T^{2} + 417 T^{3} + 3172 T^{4} + 13902 T^{5} + 3172 p T^{6} + 417 p^{2} T^{7} + 85 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 37 | \( 1 - 211 T^{2} + 22010 T^{4} - 1508006 T^{6} + 76966025 T^{8} - 3143462454 T^{10} + 76966025 p^{2} T^{12} - 1508006 p^{4} T^{14} + 22010 p^{6} T^{16} - 211 p^{8} T^{18} + p^{10} T^{20} \) |
| 41 | \( ( 1 - 2 T + 149 T^{2} - 280 T^{3} + 10370 T^{4} - 15980 T^{5} + 10370 p T^{6} - 280 p^{2} T^{7} + 149 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 43 | \( 1 - 199 T^{2} + 21890 T^{4} - 1716106 T^{6} + 103537149 T^{8} - 4965383710 T^{10} + 103537149 p^{2} T^{12} - 1716106 p^{4} T^{14} + 21890 p^{6} T^{16} - 199 p^{8} T^{18} + p^{10} T^{20} \) |
| 47 | \( 1 - 379 T^{2} + 66986 T^{4} - 7325506 T^{6} + 553060185 T^{8} - 30292826110 T^{10} + 553060185 p^{2} T^{12} - 7325506 p^{4} T^{14} + 66986 p^{6} T^{16} - 379 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( ( 1 + 11 T + 46 T^{2} - 50 T^{3} + 5265 T^{4} + 61550 T^{5} + 5265 p T^{6} - 50 p^{2} T^{7} + 46 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 59 | \( 1 - 473 T^{2} + 105113 T^{4} - 14508484 T^{6} + 1382456670 T^{8} - 95264451494 T^{10} + 1382456670 p^{2} T^{12} - 14508484 p^{4} T^{14} + 105113 p^{6} T^{16} - 473 p^{8} T^{18} + p^{10} T^{20} \) |
| 61 | \( ( 1 - 29 T + 473 T^{2} - 5144 T^{3} + 45010 T^{4} - 352102 T^{5} + 45010 p T^{6} - 5144 p^{2} T^{7} + 473 p^{3} T^{8} - 29 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 67 | \( 1 - 439 T^{2} + 96338 T^{4} - 13804762 T^{6} + 1421264109 T^{8} - 109460813470 T^{10} + 1421264109 p^{2} T^{12} - 13804762 p^{4} T^{14} + 96338 p^{6} T^{16} - 439 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( 1 - 242 T^{2} + 29477 T^{4} - 1899239 T^{6} + 58955926 T^{8} - 726732850 T^{10} + 58955926 p^{2} T^{12} - 1899239 p^{4} T^{14} + 29477 p^{6} T^{16} - 242 p^{8} T^{18} + p^{10} T^{20} \) |
| 73 | \( ( 1 - p T^{2} )^{10} \) |
| 79 | \( 1 - 622 T^{2} + 181617 T^{4} - 33011088 T^{6} + 4163684270 T^{8} - 382605651588 T^{10} + 4163684270 p^{2} T^{12} - 33011088 p^{4} T^{14} + 181617 p^{6} T^{16} - 622 p^{8} T^{18} + p^{10} T^{20} \) |
| 83 | \( ( 1 + 11 T + 348 T^{2} + 2956 T^{3} + 52851 T^{4} + 339922 T^{5} + 52851 p T^{6} + 2956 p^{2} T^{7} + 348 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 89 | \( 1 - 319 T^{2} + 57362 T^{4} - 7698494 T^{6} + 847744813 T^{8} - 79909885062 T^{10} + 847744813 p^{2} T^{12} - 7698494 p^{4} T^{14} + 57362 p^{6} T^{16} - 319 p^{8} T^{18} + p^{10} T^{20} \) |
| 97 | \( ( 1 - 13 T + 374 T^{2} - 4334 T^{3} + 67389 T^{4} - 588370 T^{5} + 67389 p T^{6} - 4334 p^{2} T^{7} + 374 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.37546697698819852795922377834, −3.37323422658574484059801579624, −3.07275346442531405988029057442, −3.05035855834055894354824217981, −3.00388970517272583245237223747, −2.79221458869641913424224807357, −2.76628242771489827756832926024, −2.65229088923241463606267925812, −2.53501556884912323141522075810, −2.39264335536283114634984865786, −2.38769726751521605960343150748, −2.31245839763432230936982294762, −2.11465883499865759327167663269, −1.84313928290178288554042513895, −1.80261609472715631062206951919, −1.65863995147113260783234368841, −1.62005418436165169266913267500, −1.61860744090438929042543835432, −1.52470215209317452496694049755, −1.36202350686145215585621055993, −1.15779880872053612004873584715, −0.843749002067110441973710676654, −0.74971201247696695777952521580, −0.54225806399376605384410922419, −0.52380866714937564919444635984,
0.52380866714937564919444635984, 0.54225806399376605384410922419, 0.74971201247696695777952521580, 0.843749002067110441973710676654, 1.15779880872053612004873584715, 1.36202350686145215585621055993, 1.52470215209317452496694049755, 1.61860744090438929042543835432, 1.62005418436165169266913267500, 1.65863995147113260783234368841, 1.80261609472715631062206951919, 1.84313928290178288554042513895, 2.11465883499865759327167663269, 2.31245839763432230936982294762, 2.38769726751521605960343150748, 2.39264335536283114634984865786, 2.53501556884912323141522075810, 2.65229088923241463606267925812, 2.76628242771489827756832926024, 2.79221458869641913424224807357, 3.00388970517272583245237223747, 3.05035855834055894354824217981, 3.07275346442531405988029057442, 3.37323422658574484059801579624, 3.37546697698819852795922377834
Plot not available for L-functions of degree greater than 10.
