| L(s) = 1 | + 2-s − 1.45·3-s + 4-s + 0.272·5-s − 1.45·6-s + 4.58·7-s + 8-s − 0.874·9-s + 0.272·10-s − 1.88·11-s − 1.45·12-s − 4.55·13-s + 4.58·14-s − 0.397·15-s + 16-s − 4.81·17-s − 0.874·18-s + 3.57·19-s + 0.272·20-s − 6.68·21-s − 1.88·22-s + 2.91·23-s − 1.45·24-s − 4.92·25-s − 4.55·26-s + 5.64·27-s + 4.58·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.841·3-s + 0.5·4-s + 0.121·5-s − 0.595·6-s + 1.73·7-s + 0.353·8-s − 0.291·9-s + 0.0862·10-s − 0.567·11-s − 0.420·12-s − 1.26·13-s + 1.22·14-s − 0.102·15-s + 0.250·16-s − 1.16·17-s − 0.206·18-s + 0.820·19-s + 0.0609·20-s − 1.45·21-s − 0.401·22-s + 0.607·23-s − 0.297·24-s − 0.985·25-s − 0.893·26-s + 1.08·27-s + 0.866·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.456267883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.456267883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 1.45T + 3T^{2} \) |
| 5 | \( 1 - 0.272T + 5T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 + 4.81T + 17T^{2} \) |
| 19 | \( 1 - 3.57T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 - 6.32T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 - 3.62T + 41T^{2} \) |
| 43 | \( 1 - 3.58T + 43T^{2} \) |
| 47 | \( 1 - 3.64T + 47T^{2} \) |
| 53 | \( 1 + 0.684T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 0.692T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 - 7.13T + 73T^{2} \) |
| 79 | \( 1 - 8.92T + 79T^{2} \) |
| 83 | \( 1 - 6.97T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−8.176082254217702958066398309779, −7.67292712575915271408882843198, −6.88440588252104310158794078581, −5.99744201193746345378578592854, −5.27592067503909995603529845124, −4.81796910829650892771515742487, −4.30047510472837697999281435550, −2.76711552127184133844232567148, −2.16835697041698730793100271377, −0.843677871608540249898667295627,
0.843677871608540249898667295627, 2.16835697041698730793100271377, 2.76711552127184133844232567148, 4.30047510472837697999281435550, 4.81796910829650892771515742487, 5.27592067503909995603529845124, 5.99744201193746345378578592854, 6.88440588252104310158794078581, 7.67292712575915271408882843198, 8.176082254217702958066398309779
