| L(s) = 1 | − 2.46·2-s − 1.92·4-s − 19.2·7-s + 24.4·8-s + 39.8·11-s + 1.00·13-s + 47.4·14-s − 44.8·16-s + 52.6·17-s − 49.5·19-s − 98.1·22-s + 27.4·23-s − 2.48·26-s + 37.1·28-s − 254.·29-s − 168.·31-s − 85.2·32-s − 129.·34-s − 419.·37-s + 122.·38-s + 398.·41-s + 358.·43-s − 76.8·44-s − 67.5·46-s + 141.·47-s + 27.1·49-s − 1.94·52-s + ⋯ |
| L(s) = 1 | − 0.871·2-s − 0.241·4-s − 1.03·7-s + 1.08·8-s + 1.09·11-s + 0.0215·13-s + 0.904·14-s − 0.700·16-s + 0.750·17-s − 0.598·19-s − 0.951·22-s + 0.248·23-s − 0.0187·26-s + 0.250·28-s − 1.63·29-s − 0.977·31-s − 0.470·32-s − 0.653·34-s − 1.86·37-s + 0.521·38-s + 1.51·41-s + 1.27·43-s − 0.263·44-s − 0.216·46-s + 0.438·47-s + 0.0792·49-s − 0.00519·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.46T + 8T^{2} \) |
| 7 | \( 1 + 19.2T + 343T^{2} \) |
| 11 | \( 1 - 39.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.00T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 27.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 419.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 398.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 358.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 290.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 28.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 732.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 176.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 802.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 512.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 612.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 80.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 24.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−8.731221171581268668889664226321, −7.58724801377701528016916880769, −7.08619552602457123502233448608, −6.12857088796005941089062106986, −5.29777075769963748271853839064, −4.03769231792029038535950663614, −3.55342701116802880321805743148, −2.09346905993720968761119938402, −1.00451559682531319899016030355, 0,
1.00451559682531319899016030355, 2.09346905993720968761119938402, 3.55342701116802880321805743148, 4.03769231792029038535950663614, 5.29777075769963748271853839064, 6.12857088796005941089062106986, 7.08619552602457123502233448608, 7.58724801377701528016916880769, 8.731221171581268668889664226321
