| L(s) = 1 | − 10.6·2-s + 22.9·3-s + 82.1·4-s + 25·5-s − 245.·6-s + 48.0·7-s − 535.·8-s + 284.·9-s − 267.·10-s − 121·11-s + 1.88e3·12-s + 1.19e3·13-s − 513.·14-s + 574.·15-s + 3.09e3·16-s − 718.·17-s − 3.04e3·18-s + 361·19-s + 2.05e3·20-s + 1.10e3·21-s + 1.29e3·22-s − 4.60e3·23-s − 1.23e4·24-s + 625·25-s − 1.27e4·26-s + 961.·27-s + 3.94e3·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 1.47·3-s + 2.56·4-s + 0.447·5-s − 2.78·6-s + 0.370·7-s − 2.95·8-s + 1.17·9-s − 0.844·10-s − 0.301·11-s + 3.78·12-s + 1.96·13-s − 0.700·14-s + 0.659·15-s + 3.02·16-s − 0.603·17-s − 2.21·18-s + 0.229·19-s + 1.14·20-s + 0.546·21-s + 0.569·22-s − 1.81·23-s − 4.36·24-s + 0.200·25-s − 3.70·26-s + 0.253·27-s + 0.952·28-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 | \( 1 + 10.6T + 32T^{2} \) |
| 3 | \( 1 - 22.9T + 243T^{2} \) |
| 7 | \( 1 - 48.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.19e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 718.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.24e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.54e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.92e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.99e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.37e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 880.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.80e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.11e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.10e5T + 8.58e9T^{2} \) |
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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
−8.799892906509510995788597936076, −8.103430782715260938454313988450, −7.74990404356340870205265267784, −6.59895023870773903215846635966, −5.81759816534416684338763166000, −3.88039012490157238202550897058, −2.95682239643333381153147331578, −1.85523356007536989673219862447, −1.53365315124364321110363184281, 0,
1.53365315124364321110363184281, 1.85523356007536989673219862447, 2.95682239643333381153147331578, 3.88039012490157238202550897058, 5.81759816534416684338763166000, 6.59895023870773903215846635966, 7.74990404356340870205265267784, 8.103430782715260938454313988450, 8.799892906509510995788597936076
