| L(s) = 1 | + 8.33·2-s − 19.5·3-s + 37.4·4-s − 25·5-s − 162.·6-s + 164.·7-s + 45.3·8-s + 137.·9-s − 208.·10-s − 121·11-s − 730.·12-s − 965.·13-s + 1.36e3·14-s + 487.·15-s − 820.·16-s + 1.24e3·17-s + 1.14e3·18-s + 361·19-s − 936.·20-s − 3.20e3·21-s − 1.00e3·22-s − 2.44e3·23-s − 885.·24-s + 625·25-s − 8.04e3·26-s + 2.05e3·27-s + 6.14e3·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 1.25·3-s + 1.17·4-s − 0.447·5-s − 1.84·6-s + 1.26·7-s + 0.250·8-s + 0.567·9-s − 0.658·10-s − 0.301·11-s − 1.46·12-s − 1.58·13-s + 1.86·14-s + 0.559·15-s − 0.800·16-s + 1.04·17-s + 0.836·18-s + 0.229·19-s − 0.523·20-s − 1.58·21-s − 0.444·22-s − 0.964·23-s − 0.313·24-s + 0.200·25-s − 2.33·26-s + 0.541·27-s + 1.48·28-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
| \(L(3)\) |
\(\approx\) |
\(2.361722538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.361722538\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 8.33T + 32T^{2} \) |
| 3 | \( 1 + 19.5T + 243T^{2} \) |
| 7 | \( 1 - 164.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 965.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.24e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.44e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 896.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.33e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.93e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.34e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.90e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.21e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.50e4T + 8.58e9T^{2} \) |
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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
−9.343271403608555842255912223310, −7.85856750235597736415856204388, −7.43609204953133697209407007128, −6.18063830798625940717366304425, −5.55687636126799123996103649639, −4.81064935687542303783902667956, −4.44333889592592093162173430375, −3.13549415974313500591125765885, −1.99673841241762076517130129030, −0.54889683079507804609976800071,
0.54889683079507804609976800071, 1.99673841241762076517130129030, 3.13549415974313500591125765885, 4.44333889592592093162173430375, 4.81064935687542303783902667956, 5.55687636126799123996103649639, 6.18063830798625940717366304425, 7.43609204953133697209407007128, 7.85856750235597736415856204388, 9.343271403608555842255912223310
