| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 3·5-s − 2·6-s + 2·7-s − 4·8-s − 4·9-s + 6·10-s − 3·11-s + 3·12-s − 4·14-s − 3·15-s + 5·16-s + 6·17-s + 8·18-s − 9·20-s + 2·21-s + 6·22-s − 4·24-s − 2·25-s − 6·27-s + 6·28-s + 7·29-s + 6·30-s + 12·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 1.34·5-s − 0.816·6-s + 0.755·7-s − 1.41·8-s − 4/3·9-s + 1.89·10-s − 0.904·11-s + 0.866·12-s − 1.06·14-s − 0.774·15-s + 5/4·16-s + 1.45·17-s + 1.88·18-s − 2.01·20-s + 0.436·21-s + 1.27·22-s − 0.816·24-s − 2/5·25-s − 1.15·27-s + 1.13·28-s + 1.29·29-s + 1.09·30-s + 2.15·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.136745465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136745465\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 59 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 83 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 19 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 75 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15 T + 139 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 17 T + 167 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 23 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 153 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 242 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 209 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 21 T + 293 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 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
−8.188036296471928122640384861596, −8.162646111900772498179998260656, −7.88725589967397767720685208378, −7.61141231546193969286285753974, −7.40085483958602519693866553164, −6.73566626773931879809450467347, −6.27883078710622879192025017049, −5.99273230443321238428187674388, −5.72674116193616171270915631150, −5.04646806608112318358805159491, −4.66859573531801999687659414156, −4.47951579912647011050824795580, −3.67164167160980973347791700435, −3.27135575430449795238275921617, −2.93208619565584855613044444588, −2.78165750864897515899034714034, −2.04147363234434574697612834699, −1.62953851181612196384905410238, −0.71292337534387395828772496703, −0.52957002926456084968500642065,
0.52957002926456084968500642065, 0.71292337534387395828772496703, 1.62953851181612196384905410238, 2.04147363234434574697612834699, 2.78165750864897515899034714034, 2.93208619565584855613044444588, 3.27135575430449795238275921617, 3.67164167160980973347791700435, 4.47951579912647011050824795580, 4.66859573531801999687659414156, 5.04646806608112318358805159491, 5.72674116193616171270915631150, 5.99273230443321238428187674388, 6.27883078710622879192025017049, 6.73566626773931879809450467347, 7.40085483958602519693866553164, 7.61141231546193969286285753974, 7.88725589967397767720685208378, 8.162646111900772498179998260656, 8.188036296471928122640384861596
