| L(s) = 1 | + 2.16·2-s + 2.56·3-s − 27.2·4-s + 5.57·6-s + 75.5·7-s − 128.·8-s − 236.·9-s + 624.·11-s − 70.1·12-s + 169·13-s + 163.·14-s + 594.·16-s − 2.34e3·17-s − 512.·18-s − 283.·19-s + 194.·21-s + 1.35e3·22-s − 2.04e3·23-s − 330.·24-s + 366.·26-s − 1.23e3·27-s − 2.06e3·28-s + 6.17e3·29-s + 687.·31-s + 5.40e3·32-s + 1.60e3·33-s − 5.08e3·34-s + ⋯ |
| L(s) = 1 | + 0.383·2-s + 0.164·3-s − 0.853·4-s + 0.0631·6-s + 0.583·7-s − 0.710·8-s − 0.972·9-s + 1.55·11-s − 0.140·12-s + 0.277·13-s + 0.223·14-s + 0.580·16-s − 1.96·17-s − 0.372·18-s − 0.180·19-s + 0.0961·21-s + 0.596·22-s − 0.805·23-s − 0.117·24-s + 0.106·26-s − 0.325·27-s − 0.497·28-s + 1.36·29-s + 0.128·31-s + 0.933·32-s + 0.256·33-s − 0.754·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.985553074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.985553074\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 - 2.16T + 32T^{2} \) |
| 3 | \( 1 - 2.56T + 243T^{2} \) |
| 7 | \( 1 - 75.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 624.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 2.34e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 283.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 687.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.32e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.54e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.60e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.39e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.01e3T + 8.58e9T^{2} \) |
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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
−11.00949111236187945640598256893, −9.603271227361373751662917341526, −8.767665392166807849210134982946, −8.326854363906833199071744918109, −6.68517383539188997965962533736, −5.81703578228648408785844807952, −4.53421273073423733768006889466, −3.87562120534043960472924085283, −2.36928963599832644546840189012, −0.73793836101671324056521906019,
0.73793836101671324056521906019, 2.36928963599832644546840189012, 3.87562120534043960472924085283, 4.53421273073423733768006889466, 5.81703578228648408785844807952, 6.68517383539188997965962533736, 8.326854363906833199071744918109, 8.767665392166807849210134982946, 9.603271227361373751662917341526, 11.00949111236187945640598256893
