| L(s) = 1 | − 3.19·3-s + 8.53·5-s − 7·7-s − 16.8·9-s − 11·11-s + 68.7·13-s − 27.2·15-s − 30.5·17-s − 14.5·19-s + 22.3·21-s − 76.8·23-s − 52.2·25-s + 139.·27-s + 149.·29-s − 188.·31-s + 35.0·33-s − 59.7·35-s + 260.·37-s − 219.·39-s + 183.·41-s + 354.·43-s − 143.·45-s − 420.·47-s + 49·49-s + 97.3·51-s − 468.·53-s − 93.8·55-s + ⋯ |
| L(s) = 1 | − 0.614·3-s + 0.763·5-s − 0.377·7-s − 0.622·9-s − 0.301·11-s + 1.46·13-s − 0.468·15-s − 0.435·17-s − 0.175·19-s + 0.232·21-s − 0.696·23-s − 0.417·25-s + 0.996·27-s + 0.956·29-s − 1.09·31-s + 0.185·33-s − 0.288·35-s + 1.15·37-s − 0.900·39-s + 0.699·41-s + 1.25·43-s − 0.475·45-s − 1.30·47-s + 0.142·49-s + 0.267·51-s − 1.21·53-s − 0.230·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.537619324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537619324\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 3.19T + 27T^{2} \) |
| 5 | \( 1 - 8.53T + 125T^{2} \) |
| 13 | \( 1 - 68.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 30.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 260.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 420.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 468.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 160.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 26.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 397.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 440.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 409.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 71.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 701.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 72.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 168.T + 9.12e5T^{2} \) |
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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
−9.375902507065895758388475315025, −8.607922522827765466613865506379, −7.77288331435696071030081667520, −6.38874945053775763483164854994, −6.14520239467267956054175170763, −5.34503298762270262729692706627, −4.20207825568420452761255802731, −3.08102967890010623007957829462, −1.95905275784095163430722444713, −0.64383638573538744513967935448,
0.64383638573538744513967935448, 1.95905275784095163430722444713, 3.08102967890010623007957829462, 4.20207825568420452761255802731, 5.34503298762270262729692706627, 6.14520239467267956054175170763, 6.38874945053775763483164854994, 7.77288331435696071030081667520, 8.607922522827765466613865506379, 9.375902507065895758388475315025
