| L(s) = 1 | + 1.56·2-s + 0.438·4-s − 5-s + 5.12·7-s − 2.43·8-s − 1.56·10-s + 1.43·11-s − 2·13-s + 8·14-s − 4.68·16-s + 7.12·17-s + 5.12·19-s − 0.438·20-s + 2.24·22-s − 6.56·23-s + 25-s − 3.12·26-s + 2.24·28-s − 29-s + 4·31-s − 2.43·32-s + 11.1·34-s − 5.12·35-s − 1.68·37-s + 8·38-s + 2.43·40-s + 1.68·41-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.219·4-s − 0.447·5-s + 1.93·7-s − 0.862·8-s − 0.493·10-s + 0.433·11-s − 0.554·13-s + 2.13·14-s − 1.17·16-s + 1.72·17-s + 1.17·19-s − 0.0980·20-s + 0.478·22-s − 1.36·23-s + 0.200·25-s − 0.612·26-s + 0.424·28-s − 0.185·29-s + 0.718·31-s − 0.431·32-s + 1.90·34-s − 0.865·35-s − 0.276·37-s + 1.29·38-s + 0.385·40-s + 0.263·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.080204003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.080204003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 2.56T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 5.68T + 97T^{2} \) |
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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
−9.651664432277461380369749395776, −8.709696071474157382559872372758, −7.80880343512888857576556238814, −7.39822218725820446397430543655, −5.85732761548949942167758772339, −5.35104596423071181388411407641, −4.48827220794765230873410343403, −3.86362642456003291750199445001, −2.66775800460361348773672474845, −1.23560909985049987109707972196,
1.23560909985049987109707972196, 2.66775800460361348773672474845, 3.86362642456003291750199445001, 4.48827220794765230873410343403, 5.35104596423071181388411407641, 5.85732761548949942167758772339, 7.39822218725820446397430543655, 7.80880343512888857576556238814, 8.709696071474157382559872372758, 9.651664432277461380369749395776
