| L(s) = 1 | + 0.805·2-s − 1.35·4-s − 1.06·5-s − 0.203·7-s − 2.69·8-s − 0.857·10-s − 11-s + 0.801·13-s − 0.163·14-s + 0.530·16-s + 0.376·17-s + 2.62·19-s + 1.43·20-s − 0.805·22-s − 2.93·23-s − 3.86·25-s + 0.645·26-s + 0.274·28-s − 0.907·29-s + 2.07·31-s + 5.82·32-s + 0.303·34-s + 0.216·35-s − 2.14·37-s + 2.11·38-s + 2.87·40-s + 1.83·41-s + ⋯ |
| L(s) = 1 | + 0.569·2-s − 0.675·4-s − 0.476·5-s − 0.0767·7-s − 0.954·8-s − 0.271·10-s − 0.301·11-s + 0.222·13-s − 0.0436·14-s + 0.132·16-s + 0.0913·17-s + 0.601·19-s + 0.321·20-s − 0.171·22-s − 0.611·23-s − 0.773·25-s + 0.126·26-s + 0.0518·28-s − 0.168·29-s + 0.373·31-s + 1.02·32-s + 0.0519·34-s + 0.0365·35-s − 0.353·37-s + 0.342·38-s + 0.454·40-s + 0.286·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310138744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310138744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.805T + 2T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 + 0.203T + 7T^{2} \) |
| 13 | \( 1 - 0.801T + 13T^{2} \) |
| 17 | \( 1 - 0.376T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 + 0.907T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 - 0.685T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 67 | \( 1 + 7.59T + 67T^{2} \) |
| 71 | \( 1 - 9.23T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 - 7.41T + 79T^{2} \) |
| 83 | \( 1 - 8.55T + 83T^{2} \) |
| 89 | \( 1 - 2.97T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.970809546572430854663519717600, −7.57154886773991522889436370247, −6.40036934645768394018839938727, −5.92872124080197756718523410941, −5.00784406469487804977830653740, −4.54062673643027605180679481604, −3.58092773307889068767588234064, −3.19640891384361402164014782629, −1.92293049102032235174648026791, −0.54295278966997822666812862566,
0.54295278966997822666812862566, 1.92293049102032235174648026791, 3.19640891384361402164014782629, 3.58092773307889068767588234064, 4.54062673643027605180679481604, 5.00784406469487804977830653740, 5.92872124080197756718523410941, 6.40036934645768394018839938727, 7.57154886773991522889436370247, 7.970809546572430854663519717600
