L(s) = 1 | − 1.37·2-s − 0.103·4-s + 2.60·5-s − 2.14·7-s + 2.89·8-s − 3.58·10-s + 5.33·11-s − 0.800·13-s + 2.94·14-s − 3.78·16-s + 4.24·17-s − 3.39·19-s − 0.270·20-s − 7.34·22-s + 0.350·23-s + 1.78·25-s + 1.10·26-s + 0.222·28-s + 6.14·29-s + 1.59·31-s − 0.586·32-s − 5.85·34-s − 5.57·35-s + 1.51·37-s + 4.67·38-s + 7.54·40-s + 4.20·41-s + ⋯ |
L(s) = 1 | − 0.973·2-s − 0.0518·4-s + 1.16·5-s − 0.809·7-s + 1.02·8-s − 1.13·10-s + 1.60·11-s − 0.222·13-s + 0.787·14-s − 0.945·16-s + 1.03·17-s − 0.779·19-s − 0.0604·20-s − 1.56·22-s + 0.0730·23-s + 0.357·25-s + 0.216·26-s + 0.0419·28-s + 1.14·29-s + 0.286·31-s − 0.103·32-s − 1.00·34-s − 0.942·35-s + 0.248·37-s + 0.758·38-s + 1.19·40-s + 0.657·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.418980170\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418980170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 13 | \( 1 + 0.800T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 23 | \( 1 - 0.350T + 23T^{2} \) |
| 29 | \( 1 - 6.14T + 29T^{2} \) |
| 31 | \( 1 - 1.59T + 31T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 + 0.326T + 47T^{2} \) |
| 53 | \( 1 + 2.42T + 53T^{2} \) |
| 59 | \( 1 + 0.747T + 59T^{2} \) |
| 61 | \( 1 + 2.16T + 61T^{2} \) |
| 67 | \( 1 - 2.07T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 0.550T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 - 1.55T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 3.09T + 97T^{2} \) |
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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
−8.253453948839963244441838655618, −7.46590874507355346590696238907, −6.48266485096123777571229081896, −6.32747028877466441970923087356, −5.31832217584432810759649876671, −4.40756712346386545199176910668, −3.62908958224649587069002697231, −2.52982504705277272407050966334, −1.56651616544300235483370160767, −0.78837303159778296676396997837,
0.78837303159778296676396997837, 1.56651616544300235483370160767, 2.52982504705277272407050966334, 3.62908958224649587069002697231, 4.40756712346386545199176910668, 5.31832217584432810759649876671, 6.32747028877466441970923087356, 6.48266485096123777571229081896, 7.46590874507355346590696238907, 8.253453948839963244441838655618