| L(s) = 1 | + 6·3-s − 12·5-s − 44·7-s + 27·9-s + 52·11-s + 26·13-s − 72·15-s − 20·17-s − 60·19-s − 264·21-s + 8·23-s + 30·25-s + 108·27-s − 132·29-s + 140·31-s + 312·33-s + 528·35-s − 68·37-s + 156·39-s + 28·41-s − 324·45-s − 36·47-s + 938·49-s − 120·51-s − 668·53-s − 624·55-s − 360·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.07·5-s − 2.37·7-s + 9-s + 1.42·11-s + 0.554·13-s − 1.23·15-s − 0.285·17-s − 0.724·19-s − 2.74·21-s + 0.0725·23-s + 6/25·25-s + 0.769·27-s − 0.845·29-s + 0.811·31-s + 1.64·33-s + 2.54·35-s − 0.302·37-s + 0.640·39-s + 0.106·41-s − 1.07·45-s − 0.111·47-s + 2.73·49-s − 0.329·51-s − 1.73·53-s − 1.52·55-s − 0.836·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 44 T + 998 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 20 T + 9238 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 60 T + 10318 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T - 418 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 132 T + 28366 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 140 T + 25782 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 68 T + 68750 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 28 T + 129610 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2914 p T^{2} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 36 T + 201778 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 668 T + 406558 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 508 T + 392026 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 340 T + 471854 T^{2} + 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 940 T + 504398 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 300 T - 57006 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1124 T + 1087686 T^{2} - 1124 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1520 T + 1230686 T^{2} - 1520 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 524 T + 908810 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1900 T + 2312266 T^{2} - 1900 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1436 T + 2285142 T^{2} + 1436 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.252881379953840279705697904029, −8.181647354603333156922178649663, −7.51492828093771964763092233204, −7.42022491676070196971596302405, −6.63176087494974193241834310331, −6.57406944648552211599693786882, −6.30836090415684065497380705889, −5.96727540905980375694861088064, −5.11167173904856396059727273247, −4.60111891899007310958769048043, −4.06401210569495533277580889828, −3.86669977938203907960934271166, −3.42672279955132716746155558045, −3.25377655855136939040579983253, −2.74059425833104330764007442216, −2.15830566068087423047180991700, −1.49303663737886020414971141242, −0.936253090472063017671074681196, 0, 0,
0.936253090472063017671074681196, 1.49303663737886020414971141242, 2.15830566068087423047180991700, 2.74059425833104330764007442216, 3.25377655855136939040579983253, 3.42672279955132716746155558045, 3.86669977938203907960934271166, 4.06401210569495533277580889828, 4.60111891899007310958769048043, 5.11167173904856396059727273247, 5.96727540905980375694861088064, 6.30836090415684065497380705889, 6.57406944648552211599693786882, 6.63176087494974193241834310331, 7.42022491676070196971596302405, 7.51492828093771964763092233204, 8.181647354603333156922178649663, 8.252881379953840279705697904029
