| L(s) = 1 | + 3-s + 3.27·5-s − 7-s + 9-s − 5.27·11-s − 13-s + 3.27·15-s − 7.27·17-s − 5.27·19-s − 21-s + 5.27·23-s + 5.72·25-s + 27-s + 0.725·29-s − 8·31-s − 5.27·33-s − 3.27·35-s + 3.27·37-s − 39-s − 8.54·41-s − 5.27·43-s + 3.27·45-s + 49-s − 7.27·51-s + 10·53-s − 17.2·55-s − 5.27·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.46·5-s − 0.377·7-s + 0.333·9-s − 1.59·11-s − 0.277·13-s + 0.845·15-s − 1.76·17-s − 1.21·19-s − 0.218·21-s + 1.09·23-s + 1.14·25-s + 0.192·27-s + 0.134·29-s − 1.43·31-s − 0.918·33-s − 0.553·35-s + 0.538·37-s − 0.160·39-s − 1.33·41-s − 0.804·43-s + 0.488·45-s + 0.142·49-s − 1.01·51-s + 1.37·53-s − 2.32·55-s − 0.698·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.27T + 5T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 + 5.27T + 19T^{2} \) |
| 23 | \( 1 - 5.27T + 23T^{2} \) |
| 29 | \( 1 - 0.725T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 3.27T + 37T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 4.72T + 61T^{2} \) |
| 67 | \( 1 - 2.54T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.82T + 73T^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−8.216900614190127963971367518762, −7.07313971718622245341668269303, −6.68630345908112665857104382450, −5.73953933465557490166842826982, −5.10823540454177304436649670106, −4.31409506244711319978082290316, −3.04484818728912692464358973393, −2.38219605281060906851939569079, −1.81476262327988030858808894055, 0,
1.81476262327988030858808894055, 2.38219605281060906851939569079, 3.04484818728912692464358973393, 4.31409506244711319978082290316, 5.10823540454177304436649670106, 5.73953933465557490166842826982, 6.68630345908112665857104382450, 7.07313971718622245341668269303, 8.216900614190127963971367518762
