| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s − 5·7-s + 9-s − 2·10-s − 2·11-s + 2·12-s + 10·14-s + 15-s − 4·16-s + 2·17-s − 2·18-s + 2·20-s − 5·21-s + 4·22-s + 6·23-s + 25-s + 27-s − 10·28-s − 4·29-s − 2·30-s + 7·31-s + 8·32-s − 2·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s − 1.88·7-s + 1/3·9-s − 0.632·10-s − 0.603·11-s + 0.577·12-s + 2.67·14-s + 0.258·15-s − 16-s + 0.485·17-s − 0.471·18-s + 0.447·20-s − 1.09·21-s + 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.88·28-s − 0.742·29-s − 0.365·30-s + 1.25·31-s + 1.41·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.851454893284229582281062956712, −7.85390095983638311382121906913, −7.19501516354407225227662865619, −6.53740365023980918100365402416, −5.68617010619664460949921843815, −4.43077767969597650979846018180, −3.15121247425298096650116961286, −2.65221657319580922018121298270, −1.29741951614738079158684794030, 0,
1.29741951614738079158684794030, 2.65221657319580922018121298270, 3.15121247425298096650116961286, 4.43077767969597650979846018180, 5.68617010619664460949921843815, 6.53740365023980918100365402416, 7.19501516354407225227662865619, 7.85390095983638311382121906913, 8.851454893284229582281062956712
