| L(s) = 1 | − 2-s + 3.32·3-s + 4-s + 3.25·5-s − 3.32·6-s − 3.84·7-s − 8-s + 8.07·9-s − 3.25·10-s + 11-s + 3.32·12-s − 0.372·13-s + 3.84·14-s + 10.8·15-s + 16-s + 1.47·17-s − 8.07·18-s + 0.924·19-s + 3.25·20-s − 12.7·21-s − 22-s − 6.51·23-s − 3.32·24-s + 5.59·25-s + 0.372·26-s + 16.9·27-s − 3.84·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.92·3-s + 0.5·4-s + 1.45·5-s − 1.35·6-s − 1.45·7-s − 0.353·8-s + 2.69·9-s − 1.02·10-s + 0.301·11-s + 0.960·12-s − 0.103·13-s + 1.02·14-s + 2.79·15-s + 0.250·16-s + 0.357·17-s − 1.90·18-s + 0.212·19-s + 0.727·20-s − 2.79·21-s − 0.213·22-s − 1.35·23-s − 0.679·24-s + 1.11·25-s + 0.0729·26-s + 3.25·27-s − 0.726·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.557828729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.557828729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 3.32T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 - 0.924T + 19T^{2} \) |
| 23 | \( 1 + 6.51T + 23T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 - 0.476T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 + 3.54T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 3.63T + 97T^{2} \) |
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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
−8.540580870377290264473457817041, −7.85249903505739478798773464693, −7.02078858929958248279474984346, −6.42560817916071840281846236214, −5.74008371112161620824251895362, −4.30495466285057866358709884259, −3.41702678430394998586134560319, −2.68314741767970677937294057136, −2.17016592318111779788084720237, −1.14684852217857976708982511489,
1.14684852217857976708982511489, 2.17016592318111779788084720237, 2.68314741767970677937294057136, 3.41702678430394998586134560319, 4.30495466285057866358709884259, 5.74008371112161620824251895362, 6.42560817916071840281846236214, 7.02078858929958248279474984346, 7.85249903505739478798773464693, 8.540580870377290264473457817041
