| L(s) = 1 | − 2.27·2-s − 3-s + 3.19·4-s − 5-s + 2.27·6-s + 0.223·7-s − 2.73·8-s + 9-s + 2.27·10-s − 0.304·11-s − 3.19·12-s + 2.97·13-s − 0.508·14-s + 15-s − 0.167·16-s − 2.78·17-s − 2.27·18-s − 2.14·19-s − 3.19·20-s − 0.223·21-s + 0.695·22-s + 2.73·24-s + 25-s − 6.78·26-s − 27-s + 0.713·28-s + 4.01·29-s + ⋯ |
| L(s) = 1 | − 1.61·2-s − 0.577·3-s + 1.59·4-s − 0.447·5-s + 0.930·6-s + 0.0843·7-s − 0.965·8-s + 0.333·9-s + 0.720·10-s − 0.0919·11-s − 0.923·12-s + 0.824·13-s − 0.135·14-s + 0.258·15-s − 0.0418·16-s − 0.676·17-s − 0.537·18-s − 0.491·19-s − 0.715·20-s − 0.0487·21-s + 0.148·22-s + 0.557·24-s + 0.200·25-s − 1.32·26-s − 0.192·27-s + 0.134·28-s + 0.744·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4509302230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4509302230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 7 | \( 1 - 0.223T + 7T^{2} \) |
| 11 | \( 1 + 0.304T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 + 2.78T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 29 | \( 1 - 4.01T + 29T^{2} \) |
| 31 | \( 1 + 2.79T + 31T^{2} \) |
| 37 | \( 1 - 2.96T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 0.308T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 - 1.13T + 53T^{2} \) |
| 59 | \( 1 - 7.57T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 + 5.60T + 71T^{2} \) |
| 73 | \( 1 - 9.52T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
| 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
−8.213382841743957690221469468307, −7.16787114931413859084788263517, −6.72703044949074102265079325082, −6.11160701493112202497078422618, −5.09162406633680834879463474695, −4.32584482783204589542921636460, −3.40071435467190784287557256313, −2.26711256488387882709718513433, −1.42661631890872647663158544253, −0.46635637219992168683068128379,
0.46635637219992168683068128379, 1.42661631890872647663158544253, 2.26711256488387882709718513433, 3.40071435467190784287557256313, 4.32584482783204589542921636460, 5.09162406633680834879463474695, 6.11160701493112202497078422618, 6.72703044949074102265079325082, 7.16787114931413859084788263517, 8.213382841743957690221469468307
