| L(s) = 1 | − 2.62·2-s − 3.19·3-s + 4.89·4-s + 4.04·5-s + 8.39·6-s − 2.68·7-s − 7.61·8-s + 7.21·9-s − 10.6·10-s − 1.08·11-s − 15.6·12-s + 1.06·13-s + 7.04·14-s − 12.9·15-s + 10.1·16-s − 1.26·17-s − 18.9·18-s + 1.29·19-s + 19.7·20-s + 8.57·21-s + 2.85·22-s + 1.73·23-s + 24.3·24-s + 11.3·25-s − 2.78·26-s − 13.4·27-s − 13.1·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.84·3-s + 2.44·4-s + 1.80·5-s + 3.42·6-s − 1.01·7-s − 2.69·8-s + 2.40·9-s − 3.35·10-s − 0.328·11-s − 4.51·12-s + 0.294·13-s + 1.88·14-s − 3.33·15-s + 2.54·16-s − 0.305·17-s − 4.46·18-s + 0.296·19-s + 4.42·20-s + 1.87·21-s + 0.609·22-s + 0.362·23-s + 4.96·24-s + 2.26·25-s − 0.547·26-s − 2.59·27-s − 2.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4334895389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4334895389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + T \) |
| 163 | \( 1 - T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 3.19T + 3T^{2} \) |
| 5 | \( 1 - 4.04T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 - 1.29T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 + 5.62T + 31T^{2} \) |
| 41 | \( 1 + 7.94T + 41T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 - 5.61T + 53T^{2} \) |
| 59 | \( 1 + 5.72T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 6.39T + 71T^{2} \) |
| 73 | \( 1 + 2.28T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 0.556T + 89T^{2} \) |
| 97 | \( 1 + 7.47T + 97T^{2} \) |
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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
−8.208953120938903989100928727587, −7.00116060735474885121288338000, −6.63217593843359538695620266066, −6.34226699566493127584717741301, −5.51015439401991668483810065933, −5.02426715670823361053537591845, −3.28933700801272939033806250530, −2.16205994413363235148244209283, −1.43140389065557614588262797882, −0.53428524122555141005469697124,
0.53428524122555141005469697124, 1.43140389065557614588262797882, 2.16205994413363235148244209283, 3.28933700801272939033806250530, 5.02426715670823361053537591845, 5.51015439401991668483810065933, 6.34226699566493127584717741301, 6.63217593843359538695620266066, 7.00116060735474885121288338000, 8.208953120938903989100928727587
