| L(s) = 1 | − 2.49·2-s + 4.22·4-s + 5-s + 1.90·7-s − 5.55·8-s − 2.49·10-s − 1.06·11-s − 4.75·14-s + 5.41·16-s − 0.637·17-s + 5.73·19-s + 4.22·20-s + 2.66·22-s − 3.81·23-s + 25-s + 8.05·28-s − 9.45·29-s − 1.46·31-s − 2.40·32-s + 1.59·34-s + 1.90·35-s + 0.757·37-s − 14.3·38-s − 5.55·40-s − 0.267·41-s + 0.637·43-s − 4.52·44-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.11·4-s + 0.447·5-s + 0.720·7-s − 1.96·8-s − 0.789·10-s − 0.322·11-s − 1.27·14-s + 1.35·16-s − 0.154·17-s + 1.31·19-s + 0.945·20-s + 0.568·22-s − 0.796·23-s + 0.200·25-s + 1.52·28-s − 1.75·29-s − 0.262·31-s − 0.424·32-s + 0.272·34-s + 0.322·35-s + 0.124·37-s − 2.32·38-s − 0.878·40-s − 0.0418·41-s + 0.0971·43-s − 0.681·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9298105317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9298105317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 7 | \( 1 - 1.90T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 17 | \( 1 + 0.637T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 + 9.45T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 0.757T + 37T^{2} \) |
| 41 | \( 1 + 0.267T + 41T^{2} \) |
| 43 | \( 1 - 0.637T + 43T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 - 6.99T + 53T^{2} \) |
| 59 | \( 1 + 0.741T + 59T^{2} \) |
| 61 | \( 1 - 4.19T + 61T^{2} \) |
| 67 | \( 1 - 8.09T + 67T^{2} \) |
| 71 | \( 1 + 9.76T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 5.11T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 4.22T + 97T^{2} \) |
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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
−7.87145924951049307576173249375, −7.47202840920975035726322141271, −6.88022226098519582166630404233, −5.84403532152736117614246999415, −5.42379855949288328868716993606, −4.29065238667945428273172559728, −3.18070481855639347135578920699, −2.19953957514865454891235764994, −1.64732387994604418231175904511, −0.64218813108593808816518722240,
0.64218813108593808816518722240, 1.64732387994604418231175904511, 2.19953957514865454891235764994, 3.18070481855639347135578920699, 4.29065238667945428273172559728, 5.42379855949288328868716993606, 5.84403532152736117614246999415, 6.88022226098519582166630404233, 7.47202840920975035726322141271, 7.87145924951049307576173249375
