| L(s) = 1 | − 4·2-s + 12·4-s − 19·5-s + 14·7-s − 32·8-s + 76·10-s + 22·11-s + 49·13-s − 56·14-s + 80·16-s + 96·17-s + 271·19-s − 228·20-s − 88·22-s − 182·23-s + 69·25-s − 196·26-s + 168·28-s + 49·29-s + 28·31-s − 192·32-s − 384·34-s − 266·35-s − 453·37-s − 1.08e3·38-s + 608·40-s + 48·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.69·5-s + 0.755·7-s − 1.41·8-s + 2.40·10-s + 0.603·11-s + 1.04·13-s − 1.06·14-s + 5/4·16-s + 1.36·17-s + 3.27·19-s − 2.54·20-s − 0.852·22-s − 1.64·23-s + 0.551·25-s − 1.47·26-s + 1.13·28-s + 0.313·29-s + 0.162·31-s − 1.06·32-s − 1.93·34-s − 1.28·35-s − 2.01·37-s − 4.62·38-s + 2.40·40-s + 0.182·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.595610428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595610428\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 19 T + 292 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 49 T + 1086 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 96 T + 5182 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 271 T + 31644 T^{2} - 271 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 182 T + 23158 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 49 T + 8800 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 28 T + 52830 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 453 T + 138664 T^{2} + 453 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 48 T - 35282 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 p T + 150770 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 289 T + 92992 T^{2} + 289 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1012 T + 550702 T^{2} + 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 795 T + 339046 T^{2} + 795 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 918 T + 664450 T^{2} - 918 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 571390 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 564 T + 767554 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1013 T + 953468 T^{2} - 1013 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 350 T + 947030 T^{2} - 350 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 354 T - 494354 T^{2} + 354 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2250 T + 2632138 T^{2} - 2250 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 696 T + 519022 T^{2} + 696 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.238306946254620928103989190182, −9.151332217248115434379516064083, −8.358400803342731265662187950701, −8.150081856220365519202456657122, −7.84436858469286848299952524287, −7.61187871619662886808755147682, −7.33306980136441617519644210765, −6.80164663702185403005472917647, −6.10721157920450644466702420239, −5.93389922587223656492498060984, −5.09525343602973795729569023759, −5.02848190463321551305382745270, −3.90320161447233534781921401028, −3.84654468954613369927028091067, −3.19428603422135864228938935419, −3.05074541245347416167112410532, −1.70148025393176856208505897675, −1.62613667035334237704291847586, −0.814469457916279430669491425423, −0.52121396638043061663286847870,
0.52121396638043061663286847870, 0.814469457916279430669491425423, 1.62613667035334237704291847586, 1.70148025393176856208505897675, 3.05074541245347416167112410532, 3.19428603422135864228938935419, 3.84654468954613369927028091067, 3.90320161447233534781921401028, 5.02848190463321551305382745270, 5.09525343602973795729569023759, 5.93389922587223656492498060984, 6.10721157920450644466702420239, 6.80164663702185403005472917647, 7.33306980136441617519644210765, 7.61187871619662886808755147682, 7.84436858469286848299952524287, 8.150081856220365519202456657122, 8.358400803342731265662187950701, 9.151332217248115434379516064083, 9.238306946254620928103989190182
