| L(s) = 1 | + 6.88·3-s − 4.69·5-s − 3.62·7-s + 20.4·9-s + 43.4·11-s − 75.0·13-s − 32.3·15-s − 13.1·17-s − 19·19-s − 24.9·21-s + 96.5·23-s − 102.·25-s − 45.3·27-s − 40.5·29-s − 138.·31-s + 299.·33-s + 17.0·35-s − 348.·37-s − 516.·39-s − 44.3·41-s + 214.·43-s − 95.7·45-s + 334.·47-s − 329.·49-s − 90.5·51-s − 311.·53-s − 204.·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s − 0.419·5-s − 0.195·7-s + 0.755·9-s + 1.19·11-s − 1.60·13-s − 0.556·15-s − 0.187·17-s − 0.229·19-s − 0.259·21-s + 0.875·23-s − 0.823·25-s − 0.323·27-s − 0.259·29-s − 0.804·31-s + 1.57·33-s + 0.0822·35-s − 1.54·37-s − 2.12·39-s − 0.168·41-s + 0.760·43-s − 0.317·45-s + 1.03·47-s − 0.961·49-s − 0.248·51-s − 0.806·53-s − 0.500·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 6.88T + 27T^{2} \) |
| 5 | \( 1 + 4.69T + 125T^{2} \) |
| 7 | \( 1 + 3.62T + 343T^{2} \) |
| 11 | \( 1 - 43.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.1T + 4.91e3T^{2} \) |
| 23 | \( 1 - 96.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 40.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 348.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 44.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 311.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 502.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 54.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 312.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 642.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 837.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 349.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 521.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 63.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.098812106670112032438731153452, −8.184145668061673691158060216006, −7.36063219170986540249734023824, −6.82113161272587073328003912356, −5.47690878782251250071144128352, −4.32488119302961633864435120186, −3.58592051610840196378490488391, −2.66153050186656783266588218835, −1.68644125203223151622341295690, 0,
1.68644125203223151622341295690, 2.66153050186656783266588218835, 3.58592051610840196378490488391, 4.32488119302961633864435120186, 5.47690878782251250071144128352, 6.82113161272587073328003912356, 7.36063219170986540249734023824, 8.184145668061673691158060216006, 9.098812106670112032438731153452
