| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s − 7-s + 8-s − 2·9-s − 2·10-s − 12-s − 14-s + 2·15-s + 16-s − 2·17-s − 2·18-s + 4·19-s − 2·20-s + 21-s − 9·23-s − 24-s − 25-s + 5·27-s − 28-s + 2·30-s + 5·31-s + 32-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.288·12-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.917·19-s − 0.447·20-s + 0.218·21-s − 1.87·23-s − 0.204·24-s − 1/5·25-s + 0.962·27-s − 0.188·28-s + 0.365·30-s + 0.898·31-s + 0.176·32-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.205241198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.205241198\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.46885230376992, −12.15684138506390, −11.90180597213799, −11.52106537756146, −11.12858574175057, −10.44132786495961, −10.22698308568778, −9.581449597772368, −8.998704565174695, −8.418387482278367, −8.044079100420900, −7.471180166274088, −7.126967348119369, −6.401946717701366, −6.105689334213304, −5.605186919863294, −5.175949305140755, −4.529904707672232, −3.965875935922332, −3.754561737840898, −3.019809045815228, −2.474356529300861, −1.972739048951323, −0.8648269418158014, −0.4613856211281525,
0.4613856211281525, 0.8648269418158014, 1.972739048951323, 2.474356529300861, 3.019809045815228, 3.754561737840898, 3.965875935922332, 4.529904707672232, 5.175949305140755, 5.605186919863294, 6.105689334213304, 6.401946717701366, 7.126967348119369, 7.471180166274088, 8.044079100420900, 8.418387482278367, 8.998704565174695, 9.581449597772368, 10.22698308568778, 10.44132786495961, 11.12858574175057, 11.52106537756146, 11.90180597213799, 12.15684138506390, 12.46885230376992
