| L(s) = 1 | − 1.78·2-s + 3-s + 1.19·4-s − 1.81·5-s − 1.78·6-s − 4.42·7-s + 1.44·8-s + 9-s + 3.24·10-s + 2.59·11-s + 1.19·12-s − 13-s + 7.90·14-s − 1.81·15-s − 4.96·16-s − 5.46·17-s − 1.78·18-s − 1.26·19-s − 2.16·20-s − 4.42·21-s − 4.64·22-s + 0.290·23-s + 1.44·24-s − 1.70·25-s + 1.78·26-s + 27-s − 5.27·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.577·3-s + 0.596·4-s − 0.812·5-s − 0.729·6-s − 1.67·7-s + 0.510·8-s + 0.333·9-s + 1.02·10-s + 0.783·11-s + 0.344·12-s − 0.277·13-s + 2.11·14-s − 0.468·15-s − 1.24·16-s − 1.32·17-s − 0.421·18-s − 0.291·19-s − 0.484·20-s − 0.965·21-s − 0.990·22-s + 0.0605·23-s + 0.294·24-s − 0.340·25-s + 0.350·26-s + 0.192·27-s − 0.996·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3382157902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3382157902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.78T + 2T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 2.59T + 11T^{2} \) |
| 17 | \( 1 + 5.46T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 0.290T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 5.68T + 37T^{2} \) |
| 41 | \( 1 + 6.27T + 41T^{2} \) |
| 43 | \( 1 - 6.20T + 43T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 - 6.90T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 - 9.53T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 0.639T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 8.05T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 0.872T + 97T^{2} \) |
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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
−8.673078876454103515367960885922, −7.79946013658713043135187352203, −7.09984002455656547132327866428, −6.75033483928678403895640840667, −5.70424194425846127294007939457, −4.17705853368892522332465505378, −3.93356827434426923532018419055, −2.81551879124059960466849271998, −1.82907618886245912769183631292, −0.37808346720206575731642397783,
0.37808346720206575731642397783, 1.82907618886245912769183631292, 2.81551879124059960466849271998, 3.93356827434426923532018419055, 4.17705853368892522332465505378, 5.70424194425846127294007939457, 6.75033483928678403895640840667, 7.09984002455656547132327866428, 7.79946013658713043135187352203, 8.673078876454103515367960885922
