| L(s) = 1 | + 0.618i·3-s + 2.23·5-s + 4.85i·7-s + 2.61·9-s + 3.38·11-s + 0.381i·13-s + 1.38i·15-s + 5.85i·17-s − 6.85·19-s − 3.00·21-s − i·23-s + 5.00·25-s + 3.47i·27-s − 3.70·29-s + 8.85·31-s + ⋯ |
| L(s) = 1 | + 0.356i·3-s + 0.999·5-s + 1.83i·7-s + 0.872·9-s + 1.01·11-s + 0.105i·13-s + 0.356i·15-s + 1.41i·17-s − 1.57·19-s − 0.654·21-s − 0.208i·23-s + 1.00·25-s + 0.668i·27-s − 0.688·29-s + 1.59·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.293912792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293912792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.618iT - 3T^{2} \) |
| 7 | \( 1 - 4.85iT - 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 - 0.381iT - 13T^{2} \) |
| 17 | \( 1 - 5.85iT - 17T^{2} \) |
| 19 | \( 1 + 6.85T + 19T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 - 8.85T + 31T^{2} \) |
| 37 | \( 1 + 3.70iT - 37T^{2} \) |
| 41 | \( 1 + 3.38T + 41T^{2} \) |
| 43 | \( 1 + 6.76iT - 43T^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 - 0.763iT - 67T^{2} \) |
| 71 | \( 1 + 2.61T + 71T^{2} \) |
| 73 | \( 1 + 7.52iT - 73T^{2} \) |
| 79 | \( 1 - 5.70T + 79T^{2} \) |
| 83 | \( 1 - 5.70iT - 83T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 - 16.0iT - 97T^{2} \) |
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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
−9.277563534311712899907597715811, −8.874448414974111442674525130749, −8.205770788408444766736272458591, −6.66581620143539781578983484820, −6.29179340046815438745998539320, −5.51036736273608631414599207964, −4.58689221081780799182260439810, −3.63151586006120526617734760802, −2.24311497210516514057745001151, −1.72797408067733927151121193433,
0.894142865497532192362774522542, 1.67960189489757914005721342674, 3.04674386223366010725449921372, 4.32487028003837155697461268750, 4.64626836929443240811067997655, 6.19887520711151453972069857347, 6.68651024639409636736838918818, 7.29421285691763836100023892380, 8.132611061949263465219273902902, 9.318540514187904462146639108723
