| L(s) = 1 | − 1.19·2-s − 3.20·3-s − 0.577·4-s − 3.80·5-s + 3.81·6-s + 4.55·7-s + 3.07·8-s + 7.24·9-s + 4.54·10-s − 11-s + 1.84·12-s − 4.62·13-s − 5.42·14-s + 12.1·15-s − 2.51·16-s − 17-s − 8.64·18-s − 1.71·19-s + 2.19·20-s − 14.5·21-s + 1.19·22-s + 1.64·23-s − 9.84·24-s + 9.48·25-s + 5.52·26-s − 13.6·27-s − 2.62·28-s + ⋯ |
| L(s) = 1 | − 0.843·2-s − 1.84·3-s − 0.288·4-s − 1.70·5-s + 1.55·6-s + 1.72·7-s + 1.08·8-s + 2.41·9-s + 1.43·10-s − 0.301·11-s + 0.533·12-s − 1.28·13-s − 1.45·14-s + 3.14·15-s − 0.628·16-s − 0.242·17-s − 2.03·18-s − 0.392·19-s + 0.491·20-s − 3.17·21-s + 0.254·22-s + 0.342·23-s − 2.00·24-s + 1.89·25-s + 1.08·26-s − 2.61·27-s − 0.496·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4510219709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4510219709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 + 3.20T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 13 | \( 1 + 4.62T + 13T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 5.55T + 29T^{2} \) |
| 31 | \( 1 - 9.50T + 31T^{2} \) |
| 37 | \( 1 - 0.412T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 - 4.88T + 53T^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 - 4.61T + 61T^{2} \) |
| 67 | \( 1 - 8.61T + 67T^{2} \) |
| 71 | \( 1 - 0.0173T + 71T^{2} \) |
| 73 | \( 1 + 8.17T + 73T^{2} \) |
| 79 | \( 1 + 0.663T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.84851800374590680279217585907, −7.29544949950993287462851487789, −6.74465195751971008915832686206, −5.52108064308722051115900269819, −4.84139616964502044488516256267, −4.52461037900617853148960987365, −4.10660128053555352977992744493, −2.34473377209265616777333020721, −1.00808430781535511564391259393, −0.57376453159612041395157696036,
0.57376453159612041395157696036, 1.00808430781535511564391259393, 2.34473377209265616777333020721, 4.10660128053555352977992744493, 4.52461037900617853148960987365, 4.84139616964502044488516256267, 5.52108064308722051115900269819, 6.74465195751971008915832686206, 7.29544949950993287462851487789, 7.84851800374590680279217585907
