L(s) = 1 | + 2-s − 0.217·3-s + 4-s − 5-s − 0.217·6-s − 0.574·7-s + 8-s − 2.95·9-s − 10-s + 11-s − 0.217·12-s + 0.293·13-s − 0.574·14-s + 0.217·15-s + 16-s − 3.05·17-s − 2.95·18-s + 7.65·19-s − 20-s + 0.124·21-s + 22-s − 1.42·23-s − 0.217·24-s + 25-s + 0.293·26-s + 1.29·27-s − 0.574·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.125·3-s + 0.5·4-s − 0.447·5-s − 0.0886·6-s − 0.217·7-s + 0.353·8-s − 0.984·9-s − 0.316·10-s + 0.301·11-s − 0.0627·12-s + 0.0813·13-s − 0.153·14-s + 0.0560·15-s + 0.250·16-s − 0.739·17-s − 0.695·18-s + 1.75·19-s − 0.223·20-s + 0.0272·21-s + 0.213·22-s − 0.296·23-s − 0.0443·24-s + 0.200·25-s + 0.0575·26-s + 0.248·27-s − 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 0.217T + 3T^{2} \) |
| 7 | \( 1 + 0.574T + 7T^{2} \) |
| 13 | \( 1 - 0.293T + 13T^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 + 1.42T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 - 0.956T + 41T^{2} \) |
| 47 | \( 1 + 0.211T + 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 - 1.84T + 59T^{2} \) |
| 61 | \( 1 + 6.14T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 + 3.35T + 71T^{2} \) |
| 73 | \( 1 - 2.43T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 0.829T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.69614296907924874645092335828, −7.24221567530811379338760951541, −6.31517225643058699832838555161, −5.69478189393776072102385701449, −5.03845223497452289449099632090, −4.12715687314568132454102246478, −3.36268878475358315249909122436, −2.71512564076375267783489272858, −1.48464319226830584384081561404, 0,
1.48464319226830584384081561404, 2.71512564076375267783489272858, 3.36268878475358315249909122436, 4.12715687314568132454102246478, 5.03845223497452289449099632090, 5.69478189393776072102385701449, 6.31517225643058699832838555161, 7.24221567530811379338760951541, 7.69614296907924874645092335828