| L(s) = 1 | − 7.00·2-s − 3.95·3-s + 17.0·4-s + 25·5-s + 27.6·6-s − 243.·7-s + 104.·8-s − 227.·9-s − 175.·10-s − 121·11-s − 67.2·12-s + 1.15e3·13-s + 1.70e3·14-s − 98.8·15-s − 1.27e3·16-s − 2.26e3·17-s + 1.59e3·18-s + 361·19-s + 425.·20-s + 961.·21-s + 847.·22-s − 955.·23-s − 414.·24-s + 625·25-s − 8.11e3·26-s + 1.85e3·27-s − 4.13e3·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.253·3-s + 0.531·4-s + 0.447·5-s + 0.313·6-s − 1.87·7-s + 0.579·8-s − 0.935·9-s − 0.553·10-s − 0.301·11-s − 0.134·12-s + 1.90·13-s + 2.32·14-s − 0.113·15-s − 1.24·16-s − 1.89·17-s + 1.15·18-s + 0.229·19-s + 0.237·20-s + 0.475·21-s + 0.373·22-s − 0.376·23-s − 0.147·24-s + 0.200·25-s − 2.35·26-s + 0.491·27-s − 0.997·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 7.00T + 32T^{2} \) |
| 3 | \( 1 + 3.95T + 243T^{2} \) |
| 7 | \( 1 + 243.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.26e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 955.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.32e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 30.2T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.97e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.20e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.90e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.18e4T + 8.58e9T^{2} \) |
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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
−8.909492267269622418743821885566, −8.335255768697880960843805509492, −7.05417363826051116117129388771, −6.29846417280991152811904638208, −5.81861966231975313314489477035, −4.26868027541141043497679969848, −3.18672305771890795165081433081, −2.14917107432243905716100364094, −0.77459490512694088153368973145, 0,
0.77459490512694088153368973145, 2.14917107432243905716100364094, 3.18672305771890795165081433081, 4.26868027541141043497679969848, 5.81861966231975313314489477035, 6.29846417280991152811904638208, 7.05417363826051116117129388771, 8.335255768697880960843805509492, 8.909492267269622418743821885566
