| L(s) = 1 | + 3.18·3-s + 3.53·5-s − 7-s + 7.14·9-s − 5.11·11-s − 13-s + 11.2·15-s + 3.03·17-s − 7.46·19-s − 3.18·21-s + 1.65·23-s + 7.49·25-s + 13.2·27-s − 8.87·29-s + 0.614·31-s − 16.2·33-s − 3.53·35-s − 9.48·37-s − 3.18·39-s + 8.84·41-s + 1.42·43-s + 25.2·45-s − 3.91·47-s + 49-s + 9.67·51-s − 5.80·53-s − 18.0·55-s + ⋯ |
| L(s) = 1 | + 1.83·3-s + 1.58·5-s − 0.377·7-s + 2.38·9-s − 1.54·11-s − 0.277·13-s + 2.90·15-s + 0.736·17-s − 1.71·19-s − 0.695·21-s + 0.344·23-s + 1.49·25-s + 2.54·27-s − 1.64·29-s + 0.110·31-s − 2.83·33-s − 0.597·35-s − 1.55·37-s − 0.510·39-s + 1.38·41-s + 0.217·43-s + 3.76·45-s − 0.571·47-s + 0.142·49-s + 1.35·51-s − 0.796·53-s − 2.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.186088580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.186088580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 11 | \( 1 + 5.11T + 11T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 + 7.46T + 19T^{2} \) |
| 23 | \( 1 - 1.65T + 23T^{2} \) |
| 29 | \( 1 + 8.87T + 29T^{2} \) |
| 31 | \( 1 - 0.614T + 31T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 41 | \( 1 - 8.84T + 41T^{2} \) |
| 43 | \( 1 - 1.42T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 + 5.80T + 53T^{2} \) |
| 59 | \( 1 - 7.37T + 59T^{2} \) |
| 61 | \( 1 - 5.37T + 61T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 - 0.806T + 71T^{2} \) |
| 73 | \( 1 - 2.61T + 73T^{2} \) |
| 79 | \( 1 - 0.421T + 79T^{2} \) |
| 83 | \( 1 - 6.23T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−10.16399835474504492313744602203, −9.464688228539731374686725267711, −8.815256829991555731611047059691, −7.947844667743568092195916711332, −7.11683564303684271761613947363, −5.96890126866147002453039366919, −4.92510949204900240536993582500, −3.52647375872293050565815154145, −2.49761261456638658255812646908, −1.94794465451326583729516690032,
1.94794465451326583729516690032, 2.49761261456638658255812646908, 3.52647375872293050565815154145, 4.92510949204900240536993582500, 5.96890126866147002453039366919, 7.11683564303684271761613947363, 7.947844667743568092195916711332, 8.815256829991555731611047059691, 9.464688228539731374686725267711, 10.16399835474504492313744602203
