| L(s) = 1 | − 2-s + 4-s − 5-s + 0.942·7-s − 8-s + 10-s + 11-s + 0.239·13-s − 0.942·14-s + 16-s + 1.94·17-s − 5.66·19-s − 20-s − 22-s + 0.578·23-s + 25-s − 0.239·26-s + 0.942·28-s + 4.60·29-s − 5.46·31-s − 32-s − 1.94·34-s − 0.942·35-s − 8.54·37-s + 5.66·38-s + 40-s + 8.70·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.356·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.0663·13-s − 0.251·14-s + 0.250·16-s + 0.471·17-s − 1.29·19-s − 0.223·20-s − 0.213·22-s + 0.120·23-s + 0.200·25-s − 0.0468·26-s + 0.178·28-s + 0.854·29-s − 0.981·31-s − 0.176·32-s − 0.333·34-s − 0.159·35-s − 1.40·37-s + 0.918·38-s + 0.158·40-s + 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 0.942T + 7T^{2} \) |
| 13 | \( 1 - 0.239T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 - 0.578T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 8.54T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 - 7.08T + 43T^{2} \) |
| 47 | \( 1 + 4.64T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 6.18T + 59T^{2} \) |
| 61 | \( 1 + 2.28T + 61T^{2} \) |
| 67 | \( 1 + 9.12T + 67T^{2} \) |
| 71 | \( 1 + 9.77T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 + 6.24T + 89T^{2} \) |
| 97 | \( 1 - 2.58T + 97T^{2} \) |
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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
−7.54388464768119340224008708046, −6.84524174060554971424721694392, −6.19596156890059037953175712850, −5.41775647723627944463362819343, −4.52469985833939841038205357825, −3.84660142829960184861932242791, −2.95891906392826360277653085352, −2.04206012126735740152020784735, −1.16128951797308356870347745942, 0,
1.16128951797308356870347745942, 2.04206012126735740152020784735, 2.95891906392826360277653085352, 3.84660142829960184861932242791, 4.52469985833939841038205357825, 5.41775647723627944463362819343, 6.19596156890059037953175712850, 6.84524174060554971424721694392, 7.54388464768119340224008708046
