| L(s) = 1 | − 4.04·2-s + 6.52·3-s + 8.39·4-s − 26.4·6-s + 7·7-s − 1.58·8-s + 15.6·9-s + 6.78·11-s + 54.7·12-s + 48.9·13-s − 28.3·14-s − 60.7·16-s + 92.4·17-s − 63.1·18-s − 125.·19-s + 45.6·21-s − 27.4·22-s − 32.2·23-s − 10.3·24-s − 198.·26-s − 74.3·27-s + 58.7·28-s + 282.·29-s + 205.·31-s + 258.·32-s + 44.2·33-s − 374.·34-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.25·3-s + 1.04·4-s − 1.79·6-s + 0.377·7-s − 0.0698·8-s + 0.578·9-s + 0.185·11-s + 1.31·12-s + 1.04·13-s − 0.541·14-s − 0.948·16-s + 1.31·17-s − 0.827·18-s − 1.51·19-s + 0.474·21-s − 0.266·22-s − 0.292·23-s − 0.0877·24-s − 1.49·26-s − 0.530·27-s + 0.396·28-s + 1.81·29-s + 1.19·31-s + 1.42·32-s + 0.233·33-s − 1.88·34-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
| \(L(2)\) |
\(\approx\) |
\(1.374221903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374221903\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.04T + 8T^{2} \) |
| 3 | \( 1 - 6.52T + 27T^{2} \) |
| 11 | \( 1 - 6.78T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 190.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 123.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 35.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 419.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 0.365T + 1.48e5T^{2} \) |
| 59 | \( 1 - 328.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 515.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 828.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 496.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 701.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 199.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 194.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 137.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 220.T + 9.12e5T^{2} \) |
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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
−12.07068914254107279465033946360, −10.80711209756187649629902415935, −10.01975864931268190773745573947, −8.947803222806360092730688244120, −8.349065652984667721137459856466, −7.72282084281961216342722238759, −6.29353904550548184639376797931, −4.20456002742259995448117779175, −2.61796985321853736301317315295, −1.18872308744899667237881485842,
1.18872308744899667237881485842, 2.61796985321853736301317315295, 4.20456002742259995448117779175, 6.29353904550548184639376797931, 7.72282084281961216342722238759, 8.349065652984667721137459856466, 8.947803222806360092730688244120, 10.01975864931268190773745573947, 10.80711209756187649629902415935, 12.07068914254107279465033946360
