| L(s) = 1 | − 2.16·3-s − 4.17·5-s − 7-s + 1.70·9-s + 2.97·11-s − 4.43·13-s + 9.05·15-s + 5.05·17-s − 19-s + 2.16·21-s + 1.98·23-s + 12.4·25-s + 2.80·27-s − 1.00·29-s + 3.05·31-s − 6.44·33-s + 4.17·35-s − 10.6·37-s + 9.61·39-s − 9.85·41-s + 5.68·43-s − 7.12·45-s − 13.1·47-s + 49-s − 10.9·51-s + 10.0·53-s − 12.4·55-s + ⋯ |
| L(s) = 1 | − 1.25·3-s − 1.86·5-s − 0.377·7-s + 0.568·9-s + 0.895·11-s − 1.22·13-s + 2.33·15-s + 1.22·17-s − 0.229·19-s + 0.473·21-s + 0.413·23-s + 2.48·25-s + 0.540·27-s − 0.187·29-s + 0.548·31-s − 1.12·33-s + 0.705·35-s − 1.74·37-s + 1.54·39-s − 1.53·41-s + 0.867·43-s − 1.06·45-s − 1.91·47-s + 0.142·49-s − 1.53·51-s + 1.37·53-s − 1.67·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3061336291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3061336291\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 + 4.17T + 5T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 9.85T + 41T^{2} \) |
| 43 | \( 1 - 5.68T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 2.97T + 73T^{2} \) |
| 79 | \( 1 + 8.52T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
| 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
−7.50003713512315809412267016357, −7.11864466793522236685330400282, −6.55855045363588306235594036132, −5.63953768049422939712616523786, −4.93011773784328329825412438058, −4.40113519190492222659876012980, −3.54408350880450614561084097468, −2.97768380500426873244738915300, −1.34561557985656094445373364899, −0.31375759169221242631910114845,
0.31375759169221242631910114845, 1.34561557985656094445373364899, 2.97768380500426873244738915300, 3.54408350880450614561084097468, 4.40113519190492222659876012980, 4.93011773784328329825412438058, 5.63953768049422939712616523786, 6.55855045363588306235594036132, 7.11864466793522236685330400282, 7.50003713512315809412267016357
