| L(s) = 1 | − 10.6·2-s − 25.7·3-s + 81.4·4-s − 25·5-s + 274.·6-s − 195.·7-s − 527.·8-s + 420.·9-s + 266.·10-s − 121·11-s − 2.09e3·12-s + 131.·13-s + 2.08e3·14-s + 643.·15-s + 3.00e3·16-s − 1.42e3·17-s − 4.47e3·18-s + 361·19-s − 2.03e3·20-s + 5.03e3·21-s + 1.28e3·22-s − 243.·23-s + 1.35e4·24-s + 625·25-s − 1.39e3·26-s − 4.56e3·27-s − 1.59e4·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 1.65·3-s + 2.54·4-s − 0.447·5-s + 3.11·6-s − 1.50·7-s − 2.91·8-s + 1.72·9-s + 0.842·10-s − 0.301·11-s − 4.20·12-s + 0.215·13-s + 2.83·14-s + 0.738·15-s + 2.93·16-s − 1.19·17-s − 3.25·18-s + 0.229·19-s − 1.13·20-s + 2.49·21-s + 0.567·22-s − 0.0958·23-s + 4.81·24-s + 0.200·25-s − 0.405·26-s − 1.20·27-s − 3.84·28-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
| \(L(3)\) |
\(\approx\) |
\(0.02660792024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02660792024\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 25.7T + 243T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 131.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.42e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 243.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.35e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 422.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.74e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.27e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.50e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.85e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.33e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.77e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.03e5T + 8.58e9T^{2} \) |
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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
−9.314162867337396492603330799292, −8.544971627023955818262951311583, −7.35920668162725998922983361725, −6.74792455730562142482133856311, −6.29291179193570824735573695524, −5.29905075351166345045219337234, −3.74738282056064948611698611027, −2.44829688343660757039467670739, −1.09038570443964049064429783202, −0.12291801494052411930188009784,
0.12291801494052411930188009784, 1.09038570443964049064429783202, 2.44829688343660757039467670739, 3.74738282056064948611698611027, 5.29905075351166345045219337234, 6.29291179193570824735573695524, 6.74792455730562142482133856311, 7.35920668162725998922983361725, 8.544971627023955818262951311583, 9.314162867337396492603330799292
