| L(s) = 1 | + 5.27·2-s − 3·3-s + 19.8·4-s − 15.8·6-s − 7·7-s + 62.3·8-s + 9·9-s + 34.7·11-s − 59.4·12-s + 37.2·13-s − 36.9·14-s + 170.·16-s + 10.5·17-s + 47.4·18-s − 58.5·19-s + 21·21-s + 183.·22-s + 125.·23-s − 187.·24-s + 196.·26-s − 27·27-s − 138.·28-s − 35.4·29-s + 291.·31-s + 399.·32-s − 104.·33-s + 55.6·34-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.47·4-s − 1.07·6-s − 0.377·7-s + 2.75·8-s + 0.333·9-s + 0.952·11-s − 1.43·12-s + 0.795·13-s − 0.704·14-s + 2.66·16-s + 0.150·17-s + 0.621·18-s − 0.707·19-s + 0.218·21-s + 1.77·22-s + 1.13·23-s − 1.59·24-s + 1.48·26-s − 0.192·27-s − 0.936·28-s − 0.226·29-s + 1.69·31-s + 2.20·32-s − 0.549·33-s + 0.280·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.798605878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.798605878\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 5.27T + 8T^{2} \) |
| 11 | \( 1 - 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.80T + 7.95e4T^{2} \) |
| 47 | \( 1 + 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−10.92538593604898412146976676843, −9.941223765484648832108103082320, −8.533857991098448212242140263991, −7.07911157728994119565938410288, −6.45071386636747692698844692531, −5.77831203483640094512377103840, −4.67939615339343324894738144429, −3.90878278724342875436975416989, −2.85471925421246258284576604243, −1.32371739493209049092897499578,
1.32371739493209049092897499578, 2.85471925421246258284576604243, 3.90878278724342875436975416989, 4.67939615339343324894738144429, 5.77831203483640094512377103840, 6.45071386636747692698844692531, 7.07911157728994119565938410288, 8.533857991098448212242140263991, 9.941223765484648832108103082320, 10.92538593604898412146976676843
