| L(s) = 1 | + 1.08·2-s − 0.826·4-s + 5-s − 2.85·7-s − 3.06·8-s + 1.08·10-s − 5.20·11-s − 1.47·13-s − 3.09·14-s − 1.66·16-s − 4.97·17-s + 5.74·19-s − 0.826·20-s − 5.63·22-s + 1.03·23-s + 25-s − 1.60·26-s + 2.36·28-s + 3.74·29-s + 4.54·31-s + 4.32·32-s − 5.38·34-s − 2.85·35-s + 1.55·37-s + 6.22·38-s − 3.06·40-s − 0.0927·41-s + ⋯ |
| L(s) = 1 | + 0.765·2-s − 0.413·4-s + 0.447·5-s − 1.07·7-s − 1.08·8-s + 0.342·10-s − 1.56·11-s − 0.409·13-s − 0.826·14-s − 0.415·16-s − 1.20·17-s + 1.31·19-s − 0.184·20-s − 1.20·22-s + 0.216·23-s + 0.200·25-s − 0.313·26-s + 0.446·28-s + 0.696·29-s + 0.816·31-s + 0.764·32-s − 0.923·34-s − 0.482·35-s + 0.255·37-s + 1.00·38-s − 0.484·40-s − 0.0144·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.430233828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.430233828\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 1.08T + 2T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 - 1.03T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + 0.0927T + 41T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 - 5.50T + 53T^{2} \) |
| 59 | \( 1 + 3.08T + 59T^{2} \) |
| 61 | \( 1 + 0.0385T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 2.07T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 - 8.84T + 83T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.542513709353575600886489535008, −7.64154236862684886579059710756, −6.77731140510257308104259198234, −6.10390976100039161330338260969, −5.30117820367881722842598389123, −4.86321596049070067457503689636, −3.87176879970430847757197672757, −2.90057779148146245901737665137, −2.49839866910996065901574039838, −0.57923119791530209966855548894,
0.57923119791530209966855548894, 2.49839866910996065901574039838, 2.90057779148146245901737665137, 3.87176879970430847757197672757, 4.86321596049070067457503689636, 5.30117820367881722842598389123, 6.10390976100039161330338260969, 6.77731140510257308104259198234, 7.64154236862684886579059710756, 8.542513709353575600886489535008
