| L(s) = 1 | + 2.54·2-s − 3-s + 4.47·4-s − 5-s − 2.54·6-s − 0.866·7-s + 6.29·8-s + 9-s − 2.54·10-s − 6.32·11-s − 4.47·12-s + 5.34·13-s − 2.20·14-s + 15-s + 7.06·16-s − 3.99·17-s + 2.54·18-s + 3.64·19-s − 4.47·20-s + 0.866·21-s − 16.0·22-s − 6.29·24-s + 25-s + 13.5·26-s − 27-s − 3.87·28-s + 2.64·29-s + ⋯ |
| L(s) = 1 | + 1.79·2-s − 0.577·3-s + 2.23·4-s − 0.447·5-s − 1.03·6-s − 0.327·7-s + 2.22·8-s + 0.333·9-s − 0.804·10-s − 1.90·11-s − 1.29·12-s + 1.48·13-s − 0.589·14-s + 0.258·15-s + 1.76·16-s − 0.968·17-s + 0.599·18-s + 0.835·19-s − 1.00·20-s + 0.189·21-s − 3.43·22-s − 1.28·24-s + 0.200·25-s + 2.66·26-s − 0.192·27-s − 0.732·28-s + 0.491·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.480537526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.480537526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 7 | \( 1 + 0.866T + 7T^{2} \) |
| 11 | \( 1 + 6.32T + 11T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 17 | \( 1 + 3.99T + 17T^{2} \) |
| 19 | \( 1 - 3.64T + 19T^{2} \) |
| 29 | \( 1 - 2.64T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 8.21T + 37T^{2} \) |
| 41 | \( 1 - 9.13T + 41T^{2} \) |
| 43 | \( 1 - 7.13T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 - 2.78T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 8.41T + 67T^{2} \) |
| 71 | \( 1 - 8.96T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 3.97T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 - 3.28T + 89T^{2} \) |
| 97 | \( 1 + 9.99T + 97T^{2} \) |
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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
−7.57579599109379802017056225091, −6.88188648454815732507092689730, −6.14833267109367741616344077704, −5.65672409520312411240386486130, −5.07187131743151756904957396641, −4.31669276311744241731045278164, −3.74045784657620904647423310105, −2.87519398323990407007104455209, −2.26652917185204134950022747741, −0.804398523809600769327066181373,
0.804398523809600769327066181373, 2.26652917185204134950022747741, 2.87519398323990407007104455209, 3.74045784657620904647423310105, 4.31669276311744241731045278164, 5.07187131743151756904957396641, 5.65672409520312411240386486130, 6.14833267109367741616344077704, 6.88188648454815732507092689730, 7.57579599109379802017056225091
