| L(s) = 1 | − 1.84·2-s + 1.65·3-s + 1.40·4-s − 5-s − 3.05·6-s + 1.09·8-s − 0.261·9-s + 1.84·10-s + 3.96·11-s + 2.32·12-s + 0.869·13-s − 1.65·15-s − 4.83·16-s + 6.02·17-s + 0.483·18-s − 2.90·19-s − 1.40·20-s − 7.32·22-s + 7.31·23-s + 1.81·24-s + 25-s − 1.60·26-s − 5.39·27-s − 1.33·29-s + 3.05·30-s + 4.18·31-s + 6.73·32-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.955·3-s + 0.703·4-s − 0.447·5-s − 1.24·6-s + 0.387·8-s − 0.0873·9-s + 0.583·10-s + 1.19·11-s + 0.671·12-s + 0.241·13-s − 0.427·15-s − 1.20·16-s + 1.46·17-s + 0.113·18-s − 0.666·19-s − 0.314·20-s − 1.56·22-s + 1.52·23-s + 0.370·24-s + 0.200·25-s − 0.314·26-s − 1.03·27-s − 0.248·29-s + 0.557·30-s + 0.750·31-s + 1.18·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.512393989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.512393989\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 1.84T + 2T^{2} \) |
| 3 | \( 1 - 1.65T + 3T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 13 | \( 1 - 0.869T + 13T^{2} \) |
| 17 | \( 1 - 6.02T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 23 | \( 1 - 7.31T + 23T^{2} \) |
| 29 | \( 1 + 1.33T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 0.185T + 43T^{2} \) |
| 47 | \( 1 + 6.01T + 47T^{2} \) |
| 53 | \( 1 + 0.214T + 53T^{2} \) |
| 59 | \( 1 - 6.27T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 - 7.81T + 71T^{2} \) |
| 73 | \( 1 - 1.24T + 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 0.951T + 97T^{2} \) |
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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
−7.945336764799869020617845593715, −7.39775320089255066637877691258, −6.69816293659830098924196077256, −5.89688784490826574773902763909, −4.82529699600005142559461389137, −4.01158590258236799489443290982, −3.32660817534984463449002650383, −2.51975798110859568632111262859, −1.46831180894662447302982699584, −0.75899931128455576979300654191,
0.75899931128455576979300654191, 1.46831180894662447302982699584, 2.51975798110859568632111262859, 3.32660817534984463449002650383, 4.01158590258236799489443290982, 4.82529699600005142559461389137, 5.89688784490826574773902763909, 6.69816293659830098924196077256, 7.39775320089255066637877691258, 7.945336764799869020617845593715
