| L(s) = 1 | + 4.62·2-s + 8.38·3-s + 13.3·4-s + 38.7·6-s − 7·7-s + 24.9·8-s + 43.3·9-s − 30.1·11-s + 112.·12-s − 88.9·13-s − 32.3·14-s + 8.10·16-s + 4.73·17-s + 200.·18-s + 124.·19-s − 58.7·21-s − 139.·22-s − 20.2·23-s + 208.·24-s − 411.·26-s + 136.·27-s − 93.7·28-s + 134.·29-s − 2.03·31-s − 161.·32-s − 252.·33-s + 21.9·34-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 1.61·3-s + 1.67·4-s + 2.63·6-s − 0.377·7-s + 1.10·8-s + 1.60·9-s − 0.825·11-s + 2.70·12-s − 1.89·13-s − 0.617·14-s + 0.126·16-s + 0.0675·17-s + 2.62·18-s + 1.50·19-s − 0.610·21-s − 1.34·22-s − 0.183·23-s + 1.77·24-s − 3.10·26-s + 0.976·27-s − 0.632·28-s + 0.858·29-s − 0.0118·31-s − 0.893·32-s − 1.33·33-s + 0.110·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.991451977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.991451977\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 4.62T + 8T^{2} \) |
| 3 | \( 1 - 8.38T + 27T^{2} \) |
| 11 | \( 1 + 30.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.73T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 20.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.03T + 2.97e4T^{2} \) |
| 37 | \( 1 - 141.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 95.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 388.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 838.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 389.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 697.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 523.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 66.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 526.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 70.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 9.27T + 7.04e5T^{2} \) |
| 97 | \( 1 - 4.19T + 9.12e5T^{2} \) |
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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
−12.66747912724749068926902162729, −11.79893074117836167096462282323, −10.15260556784260160675500006575, −9.299588515353147946848677529713, −7.80654746831271593042501031114, −7.09604721812618398512756178251, −5.44703792870315317332294990853, −4.35182692936235663235571516144, −3.03801588927631623923327660640, −2.47221646828391902923246030735,
2.47221646828391902923246030735, 3.03801588927631623923327660640, 4.35182692936235663235571516144, 5.44703792870315317332294990853, 7.09604721812618398512756178251, 7.80654746831271593042501031114, 9.299588515353147946848677529713, 10.15260556784260160675500006575, 11.79893074117836167096462282323, 12.66747912724749068926902162729
