| L(s) = 1 | + 597.·2-s − 6.56e3·3-s + 2.25e5·4-s − 3.91e6·6-s + 1.98e7·7-s + 5.62e7·8-s + 4.30e7·9-s + 6.04e8·11-s − 1.47e9·12-s + 2.50e9·13-s + 1.18e10·14-s + 4.07e9·16-s + 8.38e9·17-s + 2.57e10·18-s − 4.21e10·19-s − 1.30e11·21-s + 3.60e11·22-s + 3.84e11·23-s − 3.69e11·24-s + 1.49e12·26-s − 2.82e11·27-s + 4.46e12·28-s − 3.28e12·29-s − 5.09e10·31-s − 4.94e12·32-s − 3.96e12·33-s + 5.00e12·34-s + ⋯ |
| L(s) = 1 | + 1.64·2-s − 0.577·3-s + 1.71·4-s − 0.952·6-s + 1.29·7-s + 1.18·8-s + 0.333·9-s + 0.850·11-s − 0.992·12-s + 0.853·13-s + 2.14·14-s + 0.237·16-s + 0.291·17-s + 0.549·18-s − 0.569·19-s − 0.750·21-s + 1.40·22-s + 1.02·23-s − 0.684·24-s + 1.40·26-s − 0.192·27-s + 2.23·28-s − 1.22·29-s − 0.0107·31-s − 0.795·32-s − 0.491·33-s + 0.480·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(7.070882558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.070882558\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 597.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 1.98e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 6.04e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.50e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 8.38e9T + 8.27e20T^{2} \) |
| 19 | \( 1 + 4.21e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 3.84e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.28e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 5.09e10T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.62e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 9.05e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.32e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.95e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 5.41e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.37e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.95e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 3.10e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 2.23e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 8.15e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.37e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.51e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.60e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 8.81e15T + 5.95e33T^{2} \) |
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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
−11.34236592401517509833496760347, −10.91796741925159727082455989548, −8.966410351374201632496808548721, −7.45818455989927370826896453740, −6.29444370400870455337720554729, −5.41348691244918681092062307784, −4.47309583764526603970355723460, −3.63892970335764956454347032190, −2.08971546994246631038572630073, −1.04173747710451349319764643796,
1.04173747710451349319764643796, 2.08971546994246631038572630073, 3.63892970335764956454347032190, 4.47309583764526603970355723460, 5.41348691244918681092062307784, 6.29444370400870455337720554729, 7.45818455989927370826896453740, 8.966410351374201632496808548721, 10.91796741925159727082455989548, 11.34236592401517509833496760347
