| L(s) = 1 | + 1.59·2-s + 1.29·3-s + 0.548·4-s + 2.06·6-s + 2.14·7-s − 2.31·8-s − 1.32·9-s + 1.42·11-s + 0.710·12-s + 0.111·13-s + 3.42·14-s − 4.79·16-s + 0.0188·17-s − 2.11·18-s + 5.78·19-s + 2.77·21-s + 2.27·22-s + 6.68·23-s − 2.99·24-s + 0.177·26-s − 5.59·27-s + 1.17·28-s − 2.60·29-s + 7.75·31-s − 3.02·32-s + 1.84·33-s + 0.0300·34-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.747·3-s + 0.274·4-s + 0.843·6-s + 0.809·7-s − 0.819·8-s − 0.441·9-s + 0.429·11-s + 0.205·12-s + 0.0308·13-s + 0.914·14-s − 1.19·16-s + 0.00456·17-s − 0.498·18-s + 1.32·19-s + 0.604·21-s + 0.484·22-s + 1.39·23-s − 0.612·24-s + 0.0348·26-s − 1.07·27-s + 0.222·28-s − 0.482·29-s + 1.39·31-s − 0.534·32-s + 0.320·33-s + 0.00515·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.799365936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.799365936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 3 | \( 1 - 1.29T + 3T^{2} \) |
| 7 | \( 1 - 2.14T + 7T^{2} \) |
| 11 | \( 1 - 1.42T + 11T^{2} \) |
| 13 | \( 1 - 0.111T + 13T^{2} \) |
| 17 | \( 1 - 0.0188T + 17T^{2} \) |
| 19 | \( 1 - 5.78T + 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 - 4.60T + 41T^{2} \) |
| 43 | \( 1 + 5.99T + 43T^{2} \) |
| 47 | \( 1 - 7.28T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 8.04T + 67T^{2} \) |
| 71 | \( 1 - 0.347T + 71T^{2} \) |
| 73 | \( 1 - 8.73T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 9.99T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 8.41T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.019493979141966758301683281740, −7.44724610372751481217049327469, −6.45431360079294940038791358424, −5.79883047604282179467461793481, −4.97243700906873288948687798934, −4.57565392807173079301535108950, −3.51495158317129822277484165797, −3.08338904919448136872038027940, −2.22291013784660246650533798188, −0.968411903908015575324512215082,
0.968411903908015575324512215082, 2.22291013784660246650533798188, 3.08338904919448136872038027940, 3.51495158317129822277484165797, 4.57565392807173079301535108950, 4.97243700906873288948687798934, 5.79883047604282179467461793481, 6.45431360079294940038791358424, 7.44724610372751481217049327469, 8.019493979141966758301683281740
