| L(s) = 1 | + 5.56·2-s + 22.9·4-s − 5·5-s + 6.05·7-s + 83.0·8-s − 27.8·10-s + 11·11-s − 4.38·13-s + 33.6·14-s + 278.·16-s + 110.·17-s − 94.2·19-s − 114.·20-s + 61.1·22-s − 15.7·23-s + 25·25-s − 24.3·26-s + 138.·28-s + 256.·29-s − 170.·31-s + 883.·32-s + 614.·34-s − 30.2·35-s − 190.·37-s − 524.·38-s − 415.·40-s − 249.·41-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 2.86·4-s − 0.447·5-s + 0.326·7-s + 3.66·8-s − 0.879·10-s + 0.301·11-s − 0.0935·13-s + 0.642·14-s + 4.34·16-s + 1.57·17-s − 1.13·19-s − 1.28·20-s + 0.592·22-s − 0.142·23-s + 0.200·25-s − 0.183·26-s + 0.936·28-s + 1.64·29-s − 0.988·31-s + 4.88·32-s + 3.10·34-s − 0.146·35-s − 0.848·37-s − 2.23·38-s − 1.64·40-s − 0.950·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.209827898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.209827898\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 5.56T + 8T^{2} \) |
| 7 | \( 1 - 6.05T + 343T^{2} \) |
| 13 | \( 1 + 4.38T + 2.19e3T^{2} \) |
| 17 | \( 1 - 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 256.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 289.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 282.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 167.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 176.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 919.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 154.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 882.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 277.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 977.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3T + 9.12e5T^{2} \) |
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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
−10.86503183965574680430097268346, −10.15496716427641513583850782918, −8.371592050391066744508078171894, −7.44708976072630064886618893379, −6.57651769381684383711052904447, −5.62813980820119889892696546579, −4.72756191208414951287870713019, −3.84655261882929185047816837620, −2.90721797027695574948610345489, −1.54011202791658875869326339474,
1.54011202791658875869326339474, 2.90721797027695574948610345489, 3.84655261882929185047816837620, 4.72756191208414951287870713019, 5.62813980820119889892696546579, 6.57651769381684383711052904447, 7.44708976072630064886618893379, 8.371592050391066744508078171894, 10.15496716427641513583850782918, 10.86503183965574680430097268346
