| L(s) = 1 | + 5.31·2-s + 1.93·3-s + 20.2·4-s + 10.3·6-s + 7·7-s + 65.0·8-s − 23.2·9-s + 25.5·11-s + 39.2·12-s − 64.1·13-s + 37.1·14-s + 183.·16-s − 27.6·17-s − 123.·18-s + 0.792·19-s + 13.5·21-s + 135.·22-s + 108.·23-s + 126.·24-s − 340.·26-s − 97.4·27-s + 141.·28-s − 234.·29-s + 129.·31-s + 455.·32-s + 49.5·33-s − 147.·34-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 0.373·3-s + 2.52·4-s + 0.701·6-s + 0.377·7-s + 2.87·8-s − 0.860·9-s + 0.700·11-s + 0.944·12-s − 1.36·13-s + 0.710·14-s + 2.86·16-s − 0.395·17-s − 1.61·18-s + 0.00956·19-s + 0.141·21-s + 1.31·22-s + 0.984·23-s + 1.07·24-s − 2.56·26-s − 0.694·27-s + 0.956·28-s − 1.49·29-s + 0.748·31-s + 2.51·32-s + 0.261·33-s − 0.742·34-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
| \(L(2)\) |
\(\approx\) |
\(5.558269722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.558269722\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.31T + 8T^{2} \) |
| 3 | \( 1 - 1.93T + 27T^{2} \) |
| 11 | \( 1 - 25.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 0.792T + 6.85e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 38.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 403.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 172.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 679.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 574.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 515.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 556.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 173.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 79.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 652.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 515.T + 9.12e5T^{2} \) |
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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
−12.33347982917192994879861594794, −11.65584461523976225293688474095, −10.80015034965693102238268039423, −9.233008820908659393501233777773, −7.72245743193950273728188066175, −6.70617159427420420140411328914, −5.51016825489828582941858973925, −4.57395809141052063824924243407, −3.29365934194548690976233143321, −2.15785001882933881199835507726,
2.15785001882933881199835507726, 3.29365934194548690976233143321, 4.57395809141052063824924243407, 5.51016825489828582941858973925, 6.70617159427420420140411328914, 7.72245743193950273728188066175, 9.233008820908659393501233777773, 10.80015034965693102238268039423, 11.65584461523976225293688474095, 12.33347982917192994879861594794
