| L(s) = 1 | + 0.00358·2-s − 9·3-s − 31.9·4-s − 98.0·5-s − 0.0322·6-s − 183.·7-s − 0.229·8-s + 81·9-s − 0.351·10-s − 236.·11-s + 287.·12-s + 433.·13-s − 0.657·14-s + 882.·15-s + 1.02e3·16-s + 3.03·17-s + 0.290·18-s + 1.56e3·19-s + 3.13e3·20-s + 1.64e3·21-s − 0.848·22-s − 2.37e3·23-s + 2.06·24-s + 6.48e3·25-s + 1.55·26-s − 729·27-s + 5.86e3·28-s + ⋯ |
| L(s) = 1 | + 0.000634·2-s − 0.577·3-s − 0.999·4-s − 1.75·5-s − 0.000366·6-s − 1.41·7-s − 0.00126·8-s + 0.333·9-s − 0.00111·10-s − 0.589·11-s + 0.577·12-s + 0.711·13-s − 0.000896·14-s + 1.01·15-s + 0.999·16-s + 0.00254·17-s + 0.000211·18-s + 0.994·19-s + 1.75·20-s + 0.816·21-s − 0.000373·22-s − 0.934·23-s + 0.000732·24-s + 2.07·25-s + 0.000450·26-s − 0.192·27-s + 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 157 | \( 1 + 2.46e4T \) |
| good | 2 | \( 1 - 0.00358T + 32T^{2} \) |
| 5 | \( 1 + 98.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 183.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 236.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 433.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 3.03T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 463.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.38e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.81e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.04e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.73e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.89e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.96e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.35e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.23e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.72e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.40e3T + 8.58e9T^{2} \) |
| 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
−9.864951465394038571624025896198, −8.800927901612274467855267201856, −7.975908675781723614130738381782, −7.09751058669942217187218039786, −5.96458066423896051728994620947, −4.85883659126343014299454369320, −3.82193218627828536199470794687, −3.27110171191040395715574080142, −0.73835816920307543382926944626, 0,
0.73835816920307543382926944626, 3.27110171191040395715574080142, 3.82193218627828536199470794687, 4.85883659126343014299454369320, 5.96458066423896051728994620947, 7.09751058669942217187218039786, 7.975908675781723614130738381782, 8.800927901612274467855267201856, 9.864951465394038571624025896198
