| L(s) = 1 | − 0.179·2-s − 3-s − 1.96·4-s + 2.20·5-s + 0.179·6-s + 3.17·7-s + 0.713·8-s + 9-s − 0.396·10-s − 11-s + 1.96·12-s + 5.02·13-s − 0.570·14-s − 2.20·15-s + 3.80·16-s + 1.08·17-s − 0.179·18-s + 3.53·19-s − 4.33·20-s − 3.17·21-s + 0.179·22-s + 1.91·23-s − 0.713·24-s − 0.139·25-s − 0.903·26-s − 27-s − 6.24·28-s + ⋯ |
| L(s) = 1 | − 0.127·2-s − 0.577·3-s − 0.983·4-s + 0.986·5-s + 0.0733·6-s + 1.19·7-s + 0.252·8-s + 0.333·9-s − 0.125·10-s − 0.301·11-s + 0.568·12-s + 1.39·13-s − 0.152·14-s − 0.569·15-s + 0.951·16-s + 0.263·17-s − 0.0423·18-s + 0.811·19-s − 0.970·20-s − 0.692·21-s + 0.0383·22-s + 0.399·23-s − 0.145·24-s − 0.0278·25-s − 0.177·26-s − 0.192·27-s − 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.640425154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.640425154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.179T + 2T^{2} \) |
| 5 | \( 1 - 2.20T + 5T^{2} \) |
| 7 | \( 1 - 3.17T + 7T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 23 | \( 1 - 1.91T + 23T^{2} \) |
| 29 | \( 1 + 5.13T + 29T^{2} \) |
| 31 | \( 1 + 0.976T + 31T^{2} \) |
| 37 | \( 1 - 7.67T + 37T^{2} \) |
| 41 | \( 1 + 2.80T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 - 5.09T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 8.89T + 59T^{2} \) |
| 67 | \( 1 + 6.12T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−9.359206041879309054439129817250, −8.280457879830595594044827013870, −7.88601091789108985287196543309, −6.67011083697862521617487908899, −5.60866300023333335283429555361, −5.36353001325209730507406088565, −4.43529904013064848517228257236, −3.45489165276503630130470856418, −1.83733426533476909757950511975, −0.993667577094741447063463848212,
0.993667577094741447063463848212, 1.83733426533476909757950511975, 3.45489165276503630130470856418, 4.43529904013064848517228257236, 5.36353001325209730507406088565, 5.60866300023333335283429555361, 6.67011083697862521617487908899, 7.88601091789108985287196543309, 8.280457879830595594044827013870, 9.359206041879309054439129817250
