| L(s) = 1 | + 2·2-s + 4·4-s + 10.3·5-s − 7·7-s + 8·8-s + 20.7·10-s − 33.1·11-s + 64.7·13-s − 14·14-s + 16·16-s + 88·17-s + 96.0·19-s + 41.5·20-s − 66.3·22-s + 68.8·23-s − 17.0·25-s + 129.·26-s − 28·28-s + 4.45·29-s − 195.·31-s + 32·32-s + 176·34-s − 72.7·35-s + 30.2·37-s + 192.·38-s + 83.0·40-s + 262.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.929·5-s − 0.377·7-s + 0.353·8-s + 0.656·10-s − 0.908·11-s + 1.38·13-s − 0.267·14-s + 0.250·16-s + 1.25·17-s + 1.16·19-s + 0.464·20-s − 0.642·22-s + 0.624·23-s − 0.136·25-s + 0.977·26-s − 0.188·28-s + 0.0284·29-s − 1.13·31-s + 0.176·32-s + 0.887·34-s − 0.351·35-s + 0.134·37-s + 0.820·38-s + 0.328·40-s + 1.00·41-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
| \(L(2)\) |
\(\approx\) |
\(3.575791281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575791281\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 10.3T + 125T^{2} \) |
| 11 | \( 1 + 33.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88T + 4.91e3T^{2} \) |
| 19 | \( 1 - 96.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 68.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 4.45T + 2.43e4T^{2} \) |
| 31 | \( 1 + 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 262.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 5.22T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 700.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 283.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 453.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 665.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 766.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.20e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 71.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 69.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 125.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 693.T + 9.12e5T^{2} \) |
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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
−10.93828293142919263931211934781, −10.12793823148896034381379731883, −9.270570880474964850400000727654, −8.018062274307394876274802317123, −6.98198192720162237136296280805, −5.73790275565546404634155132928, −5.43258546184962538808685228092, −3.76767725456172585328613809283, −2.75289131490002565404802285526, −1.26880329720766215617030813417,
1.26880329720766215617030813417, 2.75289131490002565404802285526, 3.76767725456172585328613809283, 5.43258546184962538808685228092, 5.73790275565546404634155132928, 6.98198192720162237136296280805, 8.018062274307394876274802317123, 9.270570880474964850400000727654, 10.12793823148896034381379731883, 10.93828293142919263931211934781
