| L(s) = 1 | − 2·2-s + 4·4-s + 19.8·5-s + 7·7-s − 8·8-s − 39.7·10-s + 39.7·11-s + 88.6·13-s − 14·14-s + 16·16-s − 72.7·17-s + 38·19-s + 79.4·20-s − 79.4·22-s − 7.12·23-s + 269.·25-s − 177.·26-s + 28·28-s − 257.·29-s − 48.6·31-s − 32·32-s + 145.·34-s + 139.·35-s − 343.·37-s − 76·38-s − 158.·40-s − 217.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.77·5-s + 0.377·7-s − 0.353·8-s − 1.25·10-s + 1.08·11-s + 1.89·13-s − 0.267·14-s + 0.250·16-s − 1.03·17-s + 0.458·19-s + 0.888·20-s − 0.770·22-s − 0.0646·23-s + 2.15·25-s − 1.33·26-s + 0.188·28-s − 1.64·29-s − 0.281·31-s − 0.176·32-s + 0.733·34-s + 0.671·35-s − 1.52·37-s − 0.324·38-s − 0.628·40-s − 0.826·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.246622292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.246622292\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 19.8T + 125T^{2} \) |
| 11 | \( 1 - 39.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 88.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.12T + 1.21e4T^{2} \) |
| 29 | \( 1 + 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 48.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 343.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 217.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 447.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 322.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 539.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 303.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 365.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 615.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 399.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 633.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 823.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 607.T + 9.12e5T^{2} \) |
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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
−10.87272909165420384626747465201, −9.890200770027488974262119797294, −9.031928490783841360663851372592, −8.618060775702174396960470581531, −6.99326113041470095114529474988, −6.22439013263909332193343973770, −5.43339577884650802781409396874, −3.71456681385970820033301288802, −2.02927177373824624916367917399, −1.27332390200338386967832700394,
1.27332390200338386967832700394, 2.02927177373824624916367917399, 3.71456681385970820033301288802, 5.43339577884650802781409396874, 6.22439013263909332193343973770, 6.99326113041470095114529474988, 8.618060775702174396960470581531, 9.031928490783841360663851372592, 9.890200770027488974262119797294, 10.87272909165420384626747465201
