| L(s) = 1 | + 0.0489·2-s − 0.101·3-s − 1.99·4-s + 3.29·5-s − 0.00495·6-s + 4.51·7-s − 0.195·8-s − 2.98·9-s + 0.161·10-s − 0.289·11-s + 0.202·12-s + 2.10·13-s + 0.220·14-s − 0.333·15-s + 3.98·16-s + 7.50·17-s − 0.146·18-s − 5.08·19-s − 6.57·20-s − 0.457·21-s − 0.0141·22-s − 7.05·23-s + 0.0198·24-s + 5.83·25-s + 0.103·26-s + 0.606·27-s − 9.02·28-s + ⋯ |
| L(s) = 1 | + 0.0345·2-s − 0.0584·3-s − 0.998·4-s + 1.47·5-s − 0.00202·6-s + 1.70·7-s − 0.0691·8-s − 0.996·9-s + 0.0509·10-s − 0.0871·11-s + 0.0584·12-s + 0.585·13-s + 0.0590·14-s − 0.0861·15-s + 0.996·16-s + 1.82·17-s − 0.0344·18-s − 1.16·19-s − 1.47·20-s − 0.0998·21-s − 0.00301·22-s − 1.47·23-s + 0.00404·24-s + 1.16·25-s + 0.0202·26-s + 0.116·27-s − 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.912388207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.912388207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0489T + 2T^{2} \) |
| 3 | \( 1 + 0.101T + 3T^{2} \) |
| 5 | \( 1 - 3.29T + 5T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 11 | \( 1 + 0.289T + 11T^{2} \) |
| 13 | \( 1 - 2.10T + 13T^{2} \) |
| 17 | \( 1 - 7.50T + 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 + 7.05T + 23T^{2} \) |
| 29 | \( 1 - 6.47T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 - 0.452T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 0.115T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 7.71T + 61T^{2} \) |
| 67 | \( 1 + 5.57T + 67T^{2} \) |
| 71 | \( 1 + 3.10T + 71T^{2} \) |
| 73 | \( 1 + 5.98T + 73T^{2} \) |
| 79 | \( 1 - 9.61T + 79T^{2} \) |
| 83 | \( 1 + 2.15T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 9.76T + 97T^{2} \) |
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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
−10.17912078258622655479616046117, −9.027392598121315937087618714558, −8.398873763875267747906622408817, −7.87895762270465136790817907946, −6.19437063674240239967861679433, −5.55124327415210970687625978770, −5.00496155490247069861528468449, −3.84224968185398626487262559979, −2.33745473937165881368145872912, −1.22281375582550001309877724565,
1.22281375582550001309877724565, 2.33745473937165881368145872912, 3.84224968185398626487262559979, 5.00496155490247069861528468449, 5.55124327415210970687625978770, 6.19437063674240239967861679433, 7.87895762270465136790817907946, 8.398873763875267747906622408817, 9.027392598121315937087618714558, 10.17912078258622655479616046117
