| L(s) = 1 | − 2.51·3-s + 5-s − 0.519·7-s + 3.34·9-s − 6.16·11-s − 6.51·13-s − 2.51·15-s − 2.51·17-s − 6.68·19-s + 1.30·21-s + 3.03·23-s + 25-s − 0.876·27-s − 3.03·29-s + 4.33·31-s + 15.5·33-s − 0.519·35-s + 37-s + 16.4·39-s + 12.1·41-s + 2.34·43-s + 3.34·45-s − 9.54·47-s − 6.73·49-s + 6.32·51-s − 9.20·53-s − 6.16·55-s + ⋯ |
| L(s) = 1 | − 1.45·3-s + 0.447·5-s − 0.196·7-s + 1.11·9-s − 1.85·11-s − 1.80·13-s − 0.650·15-s − 0.608·17-s − 1.53·19-s + 0.285·21-s + 0.633·23-s + 0.200·25-s − 0.168·27-s − 0.564·29-s + 0.779·31-s + 2.70·33-s − 0.0878·35-s + 0.164·37-s + 2.62·39-s + 1.89·41-s + 0.357·43-s + 0.499·45-s − 1.39·47-s − 0.961·49-s + 0.885·51-s − 1.26·53-s − 0.830·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3650263365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3650263365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.51T + 3T^{2} \) |
| 7 | \( 1 + 0.519T + 7T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 13 | \( 1 + 6.51T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 + 6.68T + 19T^{2} \) |
| 23 | \( 1 - 3.03T + 23T^{2} \) |
| 29 | \( 1 + 3.03T + 29T^{2} \) |
| 31 | \( 1 - 4.33T + 31T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 + 9.54T + 47T^{2} \) |
| 53 | \( 1 + 9.20T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 0.357T + 67T^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 0.681T + 79T^{2} \) |
| 83 | \( 1 + 6.17T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 6.34T + 97T^{2} \) |
| 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
−8.762557796348371453865588993655, −7.80082019377517035238587813149, −7.11794962912593472167053936733, −6.32107177902752890801195860847, −5.69006620924904598308902070904, −4.83424223679118025844607832366, −4.60801675891498516116129207317, −2.83055028271951690415837524828, −2.13614557423074112448705269769, −0.36915076010304794162596319227,
0.36915076010304794162596319227, 2.13614557423074112448705269769, 2.83055028271951690415837524828, 4.60801675891498516116129207317, 4.83424223679118025844607832366, 5.69006620924904598308902070904, 6.32107177902752890801195860847, 7.11794962912593472167053936733, 7.80082019377517035238587813149, 8.762557796348371453865588993655
