| L(s) = 1 | − 2.38·3-s − 5-s − 4.78·7-s + 2.69·9-s + 3.12·11-s − 2.81·13-s + 2.38·15-s + 6.37·17-s − 0.114·19-s + 11.4·21-s − 5.62·23-s + 25-s + 0.723·27-s + 2.77·29-s + 6.67·31-s − 7.44·33-s + 4.78·35-s + 37-s + 6.72·39-s − 3.12·41-s + 8.57·43-s − 2.69·45-s − 3.40·47-s + 15.8·49-s − 15.2·51-s − 10.2·53-s − 3.12·55-s + ⋯ |
| L(s) = 1 | − 1.37·3-s − 0.447·5-s − 1.80·7-s + 0.898·9-s + 0.941·11-s − 0.781·13-s + 0.616·15-s + 1.54·17-s − 0.0262·19-s + 2.49·21-s − 1.17·23-s + 0.200·25-s + 0.139·27-s + 0.515·29-s + 1.19·31-s − 1.29·33-s + 0.808·35-s + 0.164·37-s + 1.07·39-s − 0.487·41-s + 1.30·43-s − 0.402·45-s − 0.496·47-s + 2.26·49-s − 2.12·51-s − 1.40·53-s − 0.420·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.38T + 3T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 + 0.114T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 8.57T + 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 9.11T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 - 4.33T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 5.68T + 83T^{2} \) |
| 89 | \( 1 + 9.95T + 89T^{2} \) |
| 97 | \( 1 + 5.62T + 97T^{2} \) |
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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
−8.305364435361646081959876699223, −7.36371667233005678310950643472, −6.62165584669153511463230716149, −6.16646798842223823711460349217, −5.49672253798160849947206373929, −4.47293814114098997955554837725, −3.64901556236596374004241939179, −2.78435249631647625495597136897, −1.01719091867096492065771285146, 0,
1.01719091867096492065771285146, 2.78435249631647625495597136897, 3.64901556236596374004241939179, 4.47293814114098997955554837725, 5.49672253798160849947206373929, 6.16646798842223823711460349217, 6.62165584669153511463230716149, 7.36371667233005678310950643472, 8.305364435361646081959876699223
