| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s − 4·11-s − 4·13-s + 5·16-s − 8·17-s + 4·19-s − 6·20-s − 8·22-s + 8·23-s + 3·25-s − 8·26-s + 4·29-s + 6·32-s − 16·34-s − 4·37-s + 8·38-s − 8·40-s − 8·41-s − 12·43-s − 12·44-s + 16·46-s − 16·47-s + 6·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s − 1.20·11-s − 1.10·13-s + 5/4·16-s − 1.94·17-s + 0.917·19-s − 1.34·20-s − 1.70·22-s + 1.66·23-s + 3/5·25-s − 1.56·26-s + 0.742·29-s + 1.06·32-s − 2.74·34-s − 0.657·37-s + 1.29·38-s − 1.26·40-s − 1.24·41-s − 1.82·43-s − 1.80·44-s + 2.35·46-s − 2.33·47-s + 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 216 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 166 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 152 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 320 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−7.920537842815060672124733476004, −7.88476916109861610507008198953, −7.20057166622237754091645406942, −6.98989522498232408510027854182, −6.74999332068459232438314348639, −6.56151052418559675105487493157, −5.63607198433078333643610145539, −5.62822110628788573907053268399, −4.89786826831927841320854713220, −4.89740918684991644817272863999, −4.42505851695255334518408492294, −4.34797234638150743916917486959, −3.38486525651344811815259674173, −3.20105120032116783877464514057, −2.79837456880461495669087046074, −2.63773083261863422579512238550, −1.59787637385686034365065774260, −1.55763784183386075095443219975, 0, 0,
1.55763784183386075095443219975, 1.59787637385686034365065774260, 2.63773083261863422579512238550, 2.79837456880461495669087046074, 3.20105120032116783877464514057, 3.38486525651344811815259674173, 4.34797234638150743916917486959, 4.42505851695255334518408492294, 4.89740918684991644817272863999, 4.89786826831927841320854713220, 5.62822110628788573907053268399, 5.63607198433078333643610145539, 6.56151052418559675105487493157, 6.74999332068459232438314348639, 6.98989522498232408510027854182, 7.20057166622237754091645406942, 7.88476916109861610507008198953, 7.920537842815060672124733476004
