| L(s) = 1 | − 2.07·2-s − 3.70·4-s − 5·5-s − 4.66·7-s + 24.2·8-s + 10.3·10-s − 8.89·11-s + 34.6·13-s + 9.66·14-s − 20.6·16-s − 2.66·17-s + 125.·19-s + 18.5·20-s + 18.4·22-s − 131.·23-s + 25·25-s − 71.7·26-s + 17.2·28-s + 71.2·29-s + 13.4·31-s − 151.·32-s + 5.52·34-s + 23.3·35-s + 283.·37-s − 260.·38-s − 121.·40-s − 383.·41-s + ⋯ |
| L(s) = 1 | − 0.733·2-s − 0.462·4-s − 0.447·5-s − 0.251·7-s + 1.07·8-s + 0.327·10-s − 0.243·11-s + 0.738·13-s + 0.184·14-s − 0.323·16-s − 0.0380·17-s + 1.51·19-s + 0.206·20-s + 0.178·22-s − 1.19·23-s + 0.200·25-s − 0.541·26-s + 0.116·28-s + 0.455·29-s + 0.0777·31-s − 0.835·32-s + 0.0278·34-s + 0.112·35-s + 1.26·37-s − 1.11·38-s − 0.479·40-s − 1.46·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
| good | 2 | \( 1 + 2.07T + 8T^{2} \) |
| 7 | \( 1 + 4.66T + 343T^{2} \) |
| 11 | \( 1 + 8.89T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.66T + 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 71.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 13.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 383.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 339.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 78.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 254.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 32.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 186.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 406.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 966.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 276.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 178.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 806.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−10.09996129217477011642993451744, −9.570154619816500168545273907974, −8.423141740978623457166859096203, −7.922974442887118907246125175194, −6.81412029266772013229097567013, −5.50623588266367123197240110177, −4.36798851081485363189753652221, −3.24896545210859475129222194361, −1.36430257258629478525139487480, 0,
1.36430257258629478525139487480, 3.24896545210859475129222194361, 4.36798851081485363189753652221, 5.50623588266367123197240110177, 6.81412029266772013229097567013, 7.922974442887118907246125175194, 8.423141740978623457166859096203, 9.570154619816500168545273907974, 10.09996129217477011642993451744
