| L(s) = 1 | + 1.34·3-s − 3.29·5-s − 0.430·7-s − 1.20·9-s − 6.38·11-s + 5.35·13-s − 4.41·15-s − 17-s + 7.92·19-s − 0.578·21-s − 5.94·23-s + 5.84·25-s − 5.63·27-s − 7.75·29-s − 1.10·31-s − 8.56·33-s + 1.41·35-s − 3.95·37-s + 7.18·39-s + 0.289·41-s − 7.67·43-s + 3.95·45-s + 4.34·47-s − 6.81·49-s − 1.34·51-s + 3.66·53-s + 21.0·55-s + ⋯ |
| L(s) = 1 | + 0.774·3-s − 1.47·5-s − 0.162·7-s − 0.400·9-s − 1.92·11-s + 1.48·13-s − 1.14·15-s − 0.242·17-s + 1.81·19-s − 0.126·21-s − 1.23·23-s + 1.16·25-s − 1.08·27-s − 1.43·29-s − 0.198·31-s − 1.49·33-s + 0.239·35-s − 0.649·37-s + 1.15·39-s + 0.0452·41-s − 1.16·43-s + 0.589·45-s + 0.633·47-s − 0.973·49-s − 0.187·51-s + 0.503·53-s + 2.83·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.016807475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.016807475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 0.430T + 7T^{2} \) |
| 11 | \( 1 + 6.38T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 + 7.75T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 - 0.289T + 41T^{2} \) |
| 43 | \( 1 + 7.67T + 43T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 - 3.66T + 53T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 - 4.12T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 5.67T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 + 1.50T + 97T^{2} \) |
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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
−7.936137370798400910890832108563, −7.56087814634726954320289079094, −6.58190472459996623866387105506, −5.53774494585568547660007887199, −5.15953138142433686598220772328, −3.88175010845736866988634520657, −3.54735819892495233300313504180, −2.92269937592360749748117116214, −1.92672492349919152477919345939, −0.45653554695277179957182348128,
0.45653554695277179957182348128, 1.92672492349919152477919345939, 2.92269937592360749748117116214, 3.54735819892495233300313504180, 3.88175010845736866988634520657, 5.15953138142433686598220772328, 5.53774494585568547660007887199, 6.58190472459996623866387105506, 7.56087814634726954320289079094, 7.936137370798400910890832108563
