| L(s) = 1 | + 2.42·2-s + 3-s + 3.86·4-s + 2.42·6-s − 7-s + 4.52·8-s + 9-s − 3.72·11-s + 3.86·12-s − 1.00·13-s − 2.42·14-s + 3.23·16-s − 17-s + 2.42·18-s − 8.53·19-s − 21-s − 9.02·22-s − 3.33·23-s + 4.52·24-s − 2.42·26-s + 27-s − 3.86·28-s + 2.87·29-s − 7.76·31-s − 1.22·32-s − 3.72·33-s − 2.42·34-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.93·4-s + 0.989·6-s − 0.377·7-s + 1.60·8-s + 0.333·9-s − 1.12·11-s + 1.11·12-s − 0.277·13-s − 0.647·14-s + 0.808·16-s − 0.242·17-s + 0.571·18-s − 1.95·19-s − 0.218·21-s − 1.92·22-s − 0.696·23-s + 0.924·24-s − 0.475·26-s + 0.192·27-s − 0.731·28-s + 0.533·29-s − 1.39·31-s − 0.216·32-s − 0.648·33-s − 0.415·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 11 | \( 1 + 3.72T + 11T^{2} \) |
| 13 | \( 1 + 1.00T + 13T^{2} \) |
| 19 | \( 1 + 8.53T + 19T^{2} \) |
| 23 | \( 1 + 3.33T + 23T^{2} \) |
| 29 | \( 1 - 2.87T + 29T^{2} \) |
| 31 | \( 1 + 7.76T + 31T^{2} \) |
| 37 | \( 1 + 0.587T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 - 4.25T + 43T^{2} \) |
| 47 | \( 1 + 13.7T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 0.809T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 1.01T + 71T^{2} \) |
| 73 | \( 1 - 5.13T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 3.30T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.14078757689139815849836529300, −6.51227516207860386048891672712, −6.00841132766781314543817302994, −5.05844475531755496004312230255, −4.68339200945083563676525868274, −3.77312673917398412744046789978, −3.31454628616426430580761176884, −2.30287961538920183522271040423, −2.04208201400575344123476176105, 0,
2.04208201400575344123476176105, 2.30287961538920183522271040423, 3.31454628616426430580761176884, 3.77312673917398412744046789978, 4.68339200945083563676525868274, 5.05844475531755496004312230255, 6.00841132766781314543817302994, 6.51227516207860386048891672712, 7.14078757689139815849836529300
