| L(s) = 1 | + 2-s − 1.35·3-s + 4-s + 2.80·5-s − 1.35·6-s − 4.57·7-s + 8-s − 1.15·9-s + 2.80·10-s − 11-s − 1.35·12-s − 6.22·13-s − 4.57·14-s − 3.80·15-s + 16-s − 3.01·17-s − 1.15·18-s + 2.80·20-s + 6.20·21-s − 22-s + 7.65·23-s − 1.35·24-s + 2.84·25-s − 6.22·26-s + 5.64·27-s − 4.57·28-s + 5.20·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.783·3-s + 0.5·4-s + 1.25·5-s − 0.554·6-s − 1.72·7-s + 0.353·8-s − 0.385·9-s + 0.885·10-s − 0.301·11-s − 0.391·12-s − 1.72·13-s − 1.22·14-s − 0.981·15-s + 0.250·16-s − 0.731·17-s − 0.272·18-s + 0.626·20-s + 1.35·21-s − 0.213·22-s + 1.59·23-s − 0.277·24-s + 0.569·25-s − 1.22·26-s + 1.08·27-s − 0.864·28-s + 0.967·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.501449647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.501449647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 + 0.287T + 41T^{2} \) |
| 43 | \( 1 + 8.80T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 + 3.24T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 0.452T + 79T^{2} \) |
| 83 | \( 1 - 6.14T + 83T^{2} \) |
| 89 | \( 1 + 9.94T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.40443746781336380380819699924, −6.75681022404191875447377000563, −6.45098716721521792171709736010, −5.69424113233929225947488179074, −5.18323838747477789244855268949, −4.61999005723385196003913518259, −3.29336868240501289126136924529, −2.77326693599957691430189575212, −2.08253689663721159093604068489, −0.52694174490265709018471195572,
0.52694174490265709018471195572, 2.08253689663721159093604068489, 2.77326693599957691430189575212, 3.29336868240501289126136924529, 4.61999005723385196003913518259, 5.18323838747477789244855268949, 5.69424113233929225947488179074, 6.45098716721521792171709736010, 6.75681022404191875447377000563, 7.40443746781336380380819699924
