| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 3·9-s + 10-s − 4·11-s + 2·13-s + 16-s − 2·17-s + 3·18-s − 4·19-s − 20-s + 4·22-s + 25-s − 2·26-s − 6·29-s − 4·31-s − 32-s + 2·34-s − 3·36-s − 37-s + 4·38-s + 40-s − 6·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.392·26-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s − 0.164·37-s + 0.648·38-s + 0.158·40-s − 0.937·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00308385360362916301851197913, −10.09679829119498000302192573656, −8.850931562365171980673741626678, −8.312703387158874843568991991704, −7.37331018075114938881780514356, −6.21035074530515276134610094877, −5.15564100419176396227847875341, −3.56437110216151834426597896111, −2.26361843784339976106632082372, 0,
2.26361843784339976106632082372, 3.56437110216151834426597896111, 5.15564100419176396227847875341, 6.21035074530515276134610094877, 7.37331018075114938881780514356, 8.312703387158874843568991991704, 8.850931562365171980673741626678, 10.09679829119498000302192573656, 11.00308385360362916301851197913
