| L(s) = 1 | − 1.96·2-s + 3-s + 1.87·4-s + 5-s − 1.96·6-s + 0.251·8-s + 9-s − 1.96·10-s + 11-s + 1.87·12-s + 0.969·13-s + 15-s − 4.23·16-s + 3.82·17-s − 1.96·18-s + 7.52·19-s + 1.87·20-s − 1.96·22-s − 1.63·23-s + 0.251·24-s + 25-s − 1.90·26-s + 27-s − 3.40·29-s − 1.96·30-s − 1.59·31-s + 7.83·32-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.577·3-s + 0.936·4-s + 0.447·5-s − 0.803·6-s + 0.0888·8-s + 0.333·9-s − 0.622·10-s + 0.301·11-s + 0.540·12-s + 0.268·13-s + 0.258·15-s − 1.05·16-s + 0.927·17-s − 0.463·18-s + 1.72·19-s + 0.418·20-s − 0.419·22-s − 0.341·23-s + 0.0513·24-s + 0.200·25-s − 0.374·26-s + 0.192·27-s − 0.632·29-s − 0.359·30-s − 0.285·31-s + 1.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.586362617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586362617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 13 | \( 1 - 0.969T + 13T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 + 3.40T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 - 8.61T + 59T^{2} \) |
| 61 | \( 1 - 9.91T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 0.653T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 5.11T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.080012788336404856182842328922, −7.24084146477581131347789496295, −6.92845823199329281671631462262, −5.78720912826077230008811770374, −5.21985574762754588882612696776, −4.09394200002994505737923133046, −3.33175150034452183136924757830, −2.39287547708278952155816157487, −1.51717233041709599131426728304, −0.819134412621717695631481697517,
0.819134412621717695631481697517, 1.51717233041709599131426728304, 2.39287547708278952155816157487, 3.33175150034452183136924757830, 4.09394200002994505737923133046, 5.21985574762754588882612696776, 5.78720912826077230008811770374, 6.92845823199329281671631462262, 7.24084146477581131347789496295, 8.080012788336404856182842328922
