| L(s) = 1 | − 1.90·3-s + 2.90·5-s + 7-s + 0.618·9-s − 2.79·11-s + 1.95·13-s − 5.52·15-s + 17-s + 4.42·19-s − 1.90·21-s − 1.72·23-s + 3.42·25-s + 4.53·27-s − 8.15·29-s − 9.64·31-s + 5.31·33-s + 2.90·35-s + 2.61·37-s − 3.71·39-s − 1.43·41-s − 8.99·43-s + 1.79·45-s − 2.65·47-s + 49-s − 1.90·51-s + 4.50·53-s − 8.10·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s + 1.29·5-s + 0.377·7-s + 0.206·9-s − 0.842·11-s + 0.541·13-s − 1.42·15-s + 0.242·17-s + 1.01·19-s − 0.415·21-s − 0.360·23-s + 0.684·25-s + 0.871·27-s − 1.51·29-s − 1.73·31-s + 0.925·33-s + 0.490·35-s + 0.429·37-s − 0.594·39-s − 0.224·41-s − 1.37·43-s + 0.267·45-s − 0.387·47-s + 0.142·49-s − 0.266·51-s + 0.618·53-s − 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 3 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 19 | \( 1 - 4.42T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 + 8.15T + 29T^{2} \) |
| 31 | \( 1 + 9.64T + 31T^{2} \) |
| 37 | \( 1 - 2.61T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 + 8.99T + 43T^{2} \) |
| 47 | \( 1 + 2.65T + 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 8.64T + 71T^{2} \) |
| 73 | \( 1 - 2.37T + 73T^{2} \) |
| 79 | \( 1 + 8.59T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.35767724319891718726799695309, −6.73901089668441822954889250464, −5.77395585617724104144959375282, −5.53215172025410825481425463019, −5.18630049264230315156167608355, −4.03308541396408244723383671780, −3.04307197706858824747922970885, −2.04557025203933149161042945176, −1.29897629261242221516898546625, 0,
1.29897629261242221516898546625, 2.04557025203933149161042945176, 3.04307197706858824747922970885, 4.03308541396408244723383671780, 5.18630049264230315156167608355, 5.53215172025410825481425463019, 5.77395585617724104144959375282, 6.73901089668441822954889250464, 7.35767724319891718726799695309
