| L(s) = 1 | + 10.9·2-s − 8.64·3-s + 88.1·4-s + 25·5-s − 94.7·6-s − 108.·7-s + 616.·8-s − 168.·9-s + 274.·10-s + 121·11-s − 762.·12-s + 331.·13-s − 1.18e3·14-s − 216.·15-s + 3.93e3·16-s + 2.19e3·17-s − 1.84e3·18-s + 361·19-s + 2.20e3·20-s + 934.·21-s + 1.32e3·22-s − 178.·23-s − 5.32e3·24-s + 625·25-s + 3.63e3·26-s + 3.55e3·27-s − 9.52e3·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.554·3-s + 2.75·4-s + 0.447·5-s − 1.07·6-s − 0.833·7-s + 3.40·8-s − 0.692·9-s + 0.866·10-s + 0.301·11-s − 1.52·12-s + 0.544·13-s − 1.61·14-s − 0.248·15-s + 3.83·16-s + 1.84·17-s − 1.34·18-s + 0.229·19-s + 1.23·20-s + 0.462·21-s + 0.584·22-s − 0.0703·23-s − 1.88·24-s + 0.200·25-s + 1.05·26-s + 0.938·27-s − 2.29·28-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
| \(L(3)\) |
\(\approx\) |
\(8.113568927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.113568927\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 10.9T + 32T^{2} \) |
| 3 | \( 1 + 8.64T + 243T^{2} \) |
| 7 | \( 1 + 108.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 331.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.19e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 178.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.34e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.46e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.53e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.30e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.89e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.22e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.29e3T + 8.58e9T^{2} \) |
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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
−9.440756137362194242487854292745, −8.001114600223018790348748430940, −6.98215138914408028292765516247, −6.26388621993504124170074885846, −5.59176253641624459221106133286, −5.19788630978505287798230565631, −3.74481083165959384609008350179, −3.31738227473794650687737204627, −2.21702104713824457527001294750, −0.964003631000270471761658255209,
0.964003631000270471761658255209, 2.21702104713824457527001294750, 3.31738227473794650687737204627, 3.74481083165959384609008350179, 5.19788630978505287798230565631, 5.59176253641624459221106133286, 6.26388621993504124170074885846, 6.98215138914408028292765516247, 8.001114600223018790348748430940, 9.440756137362194242487854292745
