| L(s) = 1 | + 0.720·2-s − 1.89·3-s − 1.48·4-s + 2.78·5-s − 1.36·6-s + 4.20·7-s − 2.50·8-s + 0.604·9-s + 2.00·10-s + 1.38·11-s + 2.81·12-s − 3.16·13-s + 3.02·14-s − 5.29·15-s + 1.15·16-s + 3.96·17-s + 0.435·18-s + 0.297·19-s − 4.12·20-s − 7.97·21-s + 0.996·22-s − 2.69·23-s + 4.76·24-s + 2.76·25-s − 2.28·26-s + 4.54·27-s − 6.22·28-s + ⋯ |
| L(s) = 1 | + 0.509·2-s − 1.09·3-s − 0.740·4-s + 1.24·5-s − 0.558·6-s + 1.58·7-s − 0.886·8-s + 0.201·9-s + 0.635·10-s + 0.416·11-s + 0.811·12-s − 0.877·13-s + 0.809·14-s − 1.36·15-s + 0.288·16-s + 0.962·17-s + 0.102·18-s + 0.0683·19-s − 0.922·20-s − 1.74·21-s + 0.212·22-s − 0.562·23-s + 0.972·24-s + 0.553·25-s − 0.447·26-s + 0.875·27-s − 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637960402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637960402\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 - 0.720T + 2T^{2} \) |
| 3 | \( 1 + 1.89T + 3T^{2} \) |
| 5 | \( 1 - 2.78T + 5T^{2} \) |
| 7 | \( 1 - 4.20T + 7T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 3.16T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 - 0.297T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 + 2.96T + 29T^{2} \) |
| 37 | \( 1 - 9.88T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 - 8.22T + 43T^{2} \) |
| 47 | \( 1 + 3.39T + 47T^{2} \) |
| 53 | \( 1 - 5.84T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 2.13T + 61T^{2} \) |
| 67 | \( 1 + 8.86T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + 8.80T + 73T^{2} \) |
| 79 | \( 1 - 0.892T + 79T^{2} \) |
| 83 | \( 1 - 6.02T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 2.89T + 97T^{2} \) |
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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
−10.00208060334280423352594513668, −9.406145268363392515937173521113, −8.386152247916724733202327221402, −7.44505210381953248477360510691, −6.03930135469070714927915380455, −5.62780302230858008891973147991, −4.97283123748830743153091007893, −4.17765100746573886671453834996, −2.44062950681442750023656616151, −1.06162720927747640350711907851,
1.06162720927747640350711907851, 2.44062950681442750023656616151, 4.17765100746573886671453834996, 4.97283123748830743153091007893, 5.62780302230858008891973147991, 6.03930135469070714927915380455, 7.44505210381953248477360510691, 8.386152247916724733202327221402, 9.406145268363392515937173521113, 10.00208060334280423352594513668
