| L(s) = 1 | − 2.56·2-s + 4.56·4-s − 5-s − 3.12·7-s − 6.56·8-s + 2.56·10-s + 5.56·11-s − 2·13-s + 8·14-s + 7.68·16-s − 1.12·17-s − 3.12·19-s − 4.56·20-s − 14.2·22-s − 2.43·23-s + 25-s + 5.12·26-s − 14.2·28-s − 29-s + 4·31-s − 6.56·32-s + 2.87·34-s + 3.12·35-s + 10.6·37-s + 8·38-s + 6.56·40-s − 10.6·41-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 2.28·4-s − 0.447·5-s − 1.18·7-s − 2.31·8-s + 0.810·10-s + 1.67·11-s − 0.554·13-s + 2.13·14-s + 1.92·16-s − 0.272·17-s − 0.716·19-s − 1.01·20-s − 3.03·22-s − 0.508·23-s + 0.200·25-s + 1.00·26-s − 2.69·28-s − 0.185·29-s + 0.718·31-s − 1.15·32-s + 0.493·34-s + 0.527·35-s + 1.75·37-s + 1.29·38-s + 1.03·40-s − 1.66·41-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
| \(L(1)\) |
\(\approx\) |
\(0.4705740778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4705740778\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.56T + 2T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 1.56T + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
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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
−9.734799860980279499639096330989, −8.857210226163958058104389956260, −8.310415183645855532279745268019, −7.24687457357214400159316646837, −6.66057547523484338102983413680, −6.10136580954572760168751076281, −4.28577465182772932857620041190, −3.21328009972045559003412485900, −2.02181087587075838974186157154, −0.64203889538914671024816812879,
0.64203889538914671024816812879, 2.02181087587075838974186157154, 3.21328009972045559003412485900, 4.28577465182772932857620041190, 6.10136580954572760168751076281, 6.66057547523484338102983413680, 7.24687457357214400159316646837, 8.310415183645855532279745268019, 8.857210226163958058104389956260, 9.734799860980279499639096330989
