| L(s) = 1 | − 2-s − 1.32·3-s + 4-s − 0.835·5-s + 1.32·6-s − 1.21·7-s − 8-s − 1.24·9-s + 0.835·10-s − 6.01·11-s − 1.32·12-s + 2.14·13-s + 1.21·14-s + 1.10·15-s + 16-s + 2.45·17-s + 1.24·18-s + 19-s − 0.835·20-s + 1.61·21-s + 6.01·22-s + 3.47·23-s + 1.32·24-s − 4.30·25-s − 2.14·26-s + 5.62·27-s − 1.21·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.765·3-s + 0.5·4-s − 0.373·5-s + 0.541·6-s − 0.461·7-s − 0.353·8-s − 0.413·9-s + 0.264·10-s − 1.81·11-s − 0.382·12-s + 0.594·13-s + 0.326·14-s + 0.286·15-s + 0.250·16-s + 0.594·17-s + 0.292·18-s + 0.229·19-s − 0.186·20-s + 0.353·21-s + 1.28·22-s + 0.725·23-s + 0.270·24-s − 0.860·25-s − 0.420·26-s + 1.08·27-s − 0.230·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 + 0.835T + 5T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 + 6.01T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 - 1.33T + 31T^{2} \) |
| 37 | \( 1 + 0.648T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 - 0.150T + 43T^{2} \) |
| 47 | \( 1 - 5.81T + 47T^{2} \) |
| 53 | \( 1 - 8.36T + 53T^{2} \) |
| 59 | \( 1 + 1.82T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 79 | \( 1 + 3.67T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 97 | \( 1 - 0.349T + 97T^{2} \) |
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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
−7.56716563110224355513296812965, −6.93001822442514172108418313199, −5.97520694199617106916117727762, −5.58897936799907488778830560541, −4.94250756261971489126607167239, −3.74229008477977220578508706100, −3.01231418340261791867209187471, −2.20485329686205349957785872542, −0.837326442249898648108734613283, 0,
0.837326442249898648108734613283, 2.20485329686205349957785872542, 3.01231418340261791867209187471, 3.74229008477977220578508706100, 4.94250756261971489126607167239, 5.58897936799907488778830560541, 5.97520694199617106916117727762, 6.93001822442514172108418313199, 7.56716563110224355513296812965
