| L(s) = 1 | + 2.15e16·2-s + 2.13e25·3-s + 3.03e32·4-s + 3.38e37·5-s + 4.60e41·6-s + 5.98e44·7-s + 3.04e48·8-s − 6.70e50·9-s + 7.30e53·10-s − 4.44e55·11-s + 6.48e57·12-s + 3.60e59·13-s + 1.29e61·14-s + 7.22e62·15-s + 1.64e64·16-s + 1.23e66·17-s − 1.44e67·18-s + 4.51e68·19-s + 1.02e70·20-s + 1.27e70·21-s − 9.59e71·22-s − 4.78e72·23-s + 6.50e73·24-s + 5.29e74·25-s + 7.78e75·26-s − 3.84e76·27-s + 1.81e77·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 0.636·3-s + 1.86·4-s + 1.36·5-s + 1.07·6-s + 0.366·7-s + 1.47·8-s − 0.595·9-s + 2.30·10-s − 0.857·11-s + 1.18·12-s + 0.914·13-s + 0.621·14-s + 0.867·15-s + 0.626·16-s + 1.83·17-s − 1.00·18-s + 1.74·19-s + 2.54·20-s + 0.233·21-s − 1.45·22-s − 0.672·23-s + 0.937·24-s + 0.858·25-s + 1.54·26-s − 1.01·27-s + 0.686·28-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
| \(L(54)\) |
\(\approx\) |
\(10.08239843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.08239843\) |
| \(L(\frac{109}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 2.15e16T + 1.62e32T^{2} \) |
| 3 | \( 1 - 2.13e25T + 1.12e51T^{2} \) |
| 5 | \( 1 - 3.38e37T + 6.16e74T^{2} \) |
| 7 | \( 1 - 5.98e44T + 2.66e90T^{2} \) |
| 11 | \( 1 + 4.44e55T + 2.68e111T^{2} \) |
| 13 | \( 1 - 3.60e59T + 1.55e119T^{2} \) |
| 17 | \( 1 - 1.23e66T + 4.55e131T^{2} \) |
| 19 | \( 1 - 4.51e68T + 6.70e136T^{2} \) |
| 23 | \( 1 + 4.78e72T + 5.06e145T^{2} \) |
| 29 | \( 1 + 2.12e78T + 2.99e156T^{2} \) |
| 31 | \( 1 + 4.66e79T + 3.76e159T^{2} \) |
| 37 | \( 1 - 9.87e83T + 6.27e167T^{2} \) |
| 41 | \( 1 - 8.52e85T + 3.69e172T^{2} \) |
| 43 | \( 1 - 7.86e86T + 6.04e174T^{2} \) |
| 47 | \( 1 + 4.69e88T + 8.21e178T^{2} \) |
| 53 | \( 1 + 1.82e92T + 3.14e184T^{2} \) |
| 59 | \( 1 - 7.60e94T + 3.02e189T^{2} \) |
| 61 | \( 1 - 2.39e95T + 1.07e191T^{2} \) |
| 67 | \( 1 + 2.24e97T + 2.45e195T^{2} \) |
| 71 | \( 1 - 6.82e98T + 1.21e198T^{2} \) |
| 73 | \( 1 + 6.57e99T + 2.37e199T^{2} \) |
| 79 | \( 1 - 3.41e101T + 1.11e203T^{2} \) |
| 83 | \( 1 + 6.07e102T + 2.19e205T^{2} \) |
| 89 | \( 1 + 2.61e104T + 3.84e208T^{2} \) |
| 97 | \( 1 + 1.90e106T + 3.84e212T^{2} \) |
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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
−13.93207381100195369833314354979, −12.99167847348482625831702690543, −11.36822055530630282663207191456, −9.660425767167935248716225303322, −7.74539174028517372551532778603, −5.76357482135128011518788527572, −5.44796976453236099744685221447, −3.54092832235692618119050593636, −2.65416958064672393860877212481, −1.50405217199365304520646661676,
1.50405217199365304520646661676, 2.65416958064672393860877212481, 3.54092832235692618119050593636, 5.44796976453236099744685221447, 5.76357482135128011518788527572, 7.74539174028517372551532778603, 9.660425767167935248716225303322, 11.36822055530630282663207191456, 12.99167847348482625831702690543, 13.93207381100195369833314354979
