| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s − 9-s + 10-s − 3·11-s + 13-s − 14-s + 16-s + 18-s − 20-s + 3·22-s − 4·25-s − 26-s + 28-s − 5·31-s − 32-s − 35-s − 36-s + 40-s + 12·43-s − 3·44-s + 45-s + 4·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.223·20-s + 0.639·22-s − 4/5·25-s − 0.196·26-s + 0.188·28-s − 0.898·31-s − 0.176·32-s − 0.169·35-s − 1/6·36-s + 0.158·40-s + 1.82·43-s − 0.452·44-s + 0.149·45-s + 0.583·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
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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
−9.105235997330038988358911242721, −8.457267595495258960116165507774, −8.025417214456409996422924336913, −7.71616641687698609997952374716, −7.24874118933905717428007018537, −6.71488757898856253648611609263, −5.94550278776061308128201472809, −5.62528526227583471701226315662, −5.04945356349766452481440070820, −4.23906603219770708579771219978, −3.76385431988666233260220859550, −2.88678992693197523497454622040, −2.34298374581264166871234144397, −1.36426882171129229244357137364, 0,
1.36426882171129229244357137364, 2.34298374581264166871234144397, 2.88678992693197523497454622040, 3.76385431988666233260220859550, 4.23906603219770708579771219978, 5.04945356349766452481440070820, 5.62528526227583471701226315662, 5.94550278776061308128201472809, 6.71488757898856253648611609263, 7.24874118933905717428007018537, 7.71616641687698609997952374716, 8.025417214456409996422924336913, 8.457267595495258960116165507774, 9.105235997330038988358911242721
