| L(s) = 1 | − 2.41e16·2-s − 3.68e24·3-s + 4.21e32·4-s + 3.92e37·5-s + 8.89e40·6-s − 1.24e45·7-s − 6.26e48·8-s − 1.11e51·9-s − 9.47e53·10-s + 5.33e55·11-s − 1.55e57·12-s + 1.01e59·13-s + 3.01e61·14-s − 1.44e62·15-s + 8.30e64·16-s − 5.90e65·17-s + 2.69e67·18-s + 1.43e68·19-s + 1.65e70·20-s + 4.59e69·21-s − 1.28e72·22-s − 9.39e72·23-s + 2.30e73·24-s + 9.22e74·25-s − 2.44e75·26-s + 8.24e75·27-s − 5.25e77·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 0.109·3-s + 2.59·4-s + 1.58·5-s + 0.208·6-s − 0.764·7-s − 3.03·8-s − 0.987·9-s − 2.99·10-s + 1.02·11-s − 0.284·12-s + 0.256·13-s + 1.44·14-s − 0.173·15-s + 3.15·16-s − 0.874·17-s + 1.87·18-s + 0.555·19-s + 4.10·20-s + 0.0838·21-s − 1.95·22-s − 1.31·23-s + 0.332·24-s + 1.49·25-s − 0.487·26-s + 0.218·27-s − 1.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(108-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+53.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(54)\) |
\(\approx\) |
\(0.9381460945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9381460945\) |
| \(L(\frac{109}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.41e16T + 1.62e32T^{2} \) |
| 3 | \( 1 + 3.68e24T + 1.12e51T^{2} \) |
| 5 | \( 1 - 3.92e37T + 6.16e74T^{2} \) |
| 7 | \( 1 + 1.24e45T + 2.66e90T^{2} \) |
| 11 | \( 1 - 5.33e55T + 2.68e111T^{2} \) |
| 13 | \( 1 - 1.01e59T + 1.55e119T^{2} \) |
| 17 | \( 1 + 5.90e65T + 4.55e131T^{2} \) |
| 19 | \( 1 - 1.43e68T + 6.70e136T^{2} \) |
| 23 | \( 1 + 9.39e72T + 5.06e145T^{2} \) |
| 29 | \( 1 - 2.08e78T + 2.99e156T^{2} \) |
| 31 | \( 1 - 8.03e79T + 3.76e159T^{2} \) |
| 37 | \( 1 + 7.64e83T + 6.27e167T^{2} \) |
| 41 | \( 1 - 7.94e85T + 3.69e172T^{2} \) |
| 43 | \( 1 + 8.52e86T + 6.04e174T^{2} \) |
| 47 | \( 1 - 3.67e89T + 8.21e178T^{2} \) |
| 53 | \( 1 + 1.18e92T + 3.14e184T^{2} \) |
| 59 | \( 1 + 3.78e94T + 3.02e189T^{2} \) |
| 61 | \( 1 - 6.03e95T + 1.07e191T^{2} \) |
| 67 | \( 1 + 1.28e97T + 2.45e195T^{2} \) |
| 71 | \( 1 + 1.39e99T + 1.21e198T^{2} \) |
| 73 | \( 1 + 3.23e98T + 2.37e199T^{2} \) |
| 79 | \( 1 + 1.61e101T + 1.11e203T^{2} \) |
| 83 | \( 1 - 3.87e102T + 2.19e205T^{2} \) |
| 89 | \( 1 - 8.90e103T + 3.84e208T^{2} \) |
| 97 | \( 1 - 1.30e106T + 3.84e212T^{2} \) |
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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
−13.98591106628516014768277950763, −11.84359480057820829056991023389, −10.35745001343862716154022734797, −9.458686911358200742400967676702, −8.572291435723913286390801090263, −6.60610897157632389604842878123, −6.01478818615109514335226244203, −2.84761275705642239284518638591, −1.82364244676335413843629367495, −0.66690155952333327256954876059,
0.66690155952333327256954876059, 1.82364244676335413843629367495, 2.84761275705642239284518638591, 6.01478818615109514335226244203, 6.60610897157632389604842878123, 8.572291435723913286390801090263, 9.458686911358200742400967676702, 10.35745001343862716154022734797, 11.84359480057820829056991023389, 13.98591106628516014768277950763
