| L(s) = 1 | − 2.65·2-s − 0.258·3-s + 5.04·4-s − 1.45·5-s + 0.686·6-s − 0.380·7-s − 8.06·8-s − 2.93·9-s + 3.84·10-s + 3.71·11-s − 1.30·12-s − 1.35·13-s + 1.01·14-s + 0.375·15-s + 11.3·16-s + 6.71·17-s + 7.78·18-s + 6.09·19-s − 7.31·20-s + 0.0984·21-s − 9.84·22-s + 7.60·23-s + 2.08·24-s − 2.89·25-s + 3.58·26-s + 1.53·27-s − 1.91·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.149·3-s + 2.52·4-s − 0.648·5-s + 0.280·6-s − 0.143·7-s − 2.85·8-s − 0.977·9-s + 1.21·10-s + 1.11·11-s − 0.376·12-s − 0.374·13-s + 0.269·14-s + 0.0968·15-s + 2.83·16-s + 1.62·17-s + 1.83·18-s + 1.39·19-s − 1.63·20-s + 0.0214·21-s − 2.09·22-s + 1.58·23-s + 0.426·24-s − 0.579·25-s + 0.702·26-s + 0.295·27-s − 0.362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7449863547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7449863547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 + 0.258T + 3T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 7 | \( 1 + 0.380T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 - 2.33T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 + 9.70T + 59T^{2} \) |
| 61 | \( 1 + 5.93T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 4.18T + 83T^{2} \) |
| 89 | \( 1 - 1.66T + 89T^{2} \) |
| 97 | \( 1 - 8.53T + 97T^{2} \) |
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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
−8.316046526489725512632256935148, −7.51411556893964741204518515759, −7.31839306195901730006961861485, −6.21526445887267286143353895393, −5.78285625500776149152347522545, −4.53072114995762156043518350759, −3.10201514828723784745308358097, −2.90838465023591052455701049434, −1.29026806704519752890940621006, −0.72952483522846718893432299107,
0.72952483522846718893432299107, 1.29026806704519752890940621006, 2.90838465023591052455701049434, 3.10201514828723784745308358097, 4.53072114995762156043518350759, 5.78285625500776149152347522545, 6.21526445887267286143353895393, 7.31839306195901730006961861485, 7.51411556893964741204518515759, 8.316046526489725512632256935148
