| L(s) = 1 | + 2.05·3-s − 0.461·5-s − 7-s + 1.22·9-s + 2.48·11-s + 4.99·13-s − 0.949·15-s − 2.40·19-s − 2.05·21-s − 8.21·23-s − 4.78·25-s − 3.64·27-s − 2.53·29-s − 8.53·31-s + 5.10·33-s + 0.461·35-s − 8.37·37-s + 10.2·39-s + 11.5·41-s − 8.65·43-s − 0.567·45-s − 1.50·47-s + 49-s − 4.95·53-s − 1.14·55-s − 4.93·57-s + 8.36·59-s + ⋯ |
| L(s) = 1 | + 1.18·3-s − 0.206·5-s − 0.377·7-s + 0.409·9-s + 0.748·11-s + 1.38·13-s − 0.245·15-s − 0.550·19-s − 0.448·21-s − 1.71·23-s − 0.957·25-s − 0.700·27-s − 0.471·29-s − 1.53·31-s + 0.889·33-s + 0.0780·35-s − 1.37·37-s + 1.64·39-s + 1.79·41-s − 1.32·43-s − 0.0846·45-s − 0.219·47-s + 0.142·49-s − 0.680·53-s − 0.154·55-s − 0.654·57-s + 1.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 0.461T + 5T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 + 8.21T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 8.53T + 31T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 8.65T + 43T^{2} \) |
| 47 | \( 1 + 1.50T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 - 8.36T + 59T^{2} \) |
| 61 | \( 1 - 8.65T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 + 9.81T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 + 8.31T + 79T^{2} \) |
| 83 | \( 1 - 2.77T + 83T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−7.69180564195947207111922637838, −6.86976351674040960475674958038, −6.07709874290954081727793960383, −5.61448422673064485158241650134, −4.21250459724216156148106895773, −3.76805955168396992683499726623, −3.32288935758431418185833408640, −2.15916145173718062302953283723, −1.58987558382940729524409670318, 0,
1.58987558382940729524409670318, 2.15916145173718062302953283723, 3.32288935758431418185833408640, 3.76805955168396992683499726623, 4.21250459724216156148106895773, 5.61448422673064485158241650134, 6.07709874290954081727793960383, 6.86976351674040960475674958038, 7.69180564195947207111922637838
