| L(s) = 1 | + 2-s − 2.35·3-s + 4-s − 3.55·5-s − 2.35·6-s − 1.02·7-s + 8-s + 2.55·9-s − 3.55·10-s + 4.90·11-s − 2.35·12-s − 0.857·13-s − 1.02·14-s + 8.37·15-s + 16-s − 0.693·17-s + 2.55·18-s + 19-s − 3.55·20-s + 2.41·21-s + 4.90·22-s + 1.52·23-s − 2.35·24-s + 7.62·25-s − 0.857·26-s + 1.04·27-s − 1.02·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.36·3-s + 0.5·4-s − 1.58·5-s − 0.962·6-s − 0.386·7-s + 0.353·8-s + 0.852·9-s − 1.12·10-s + 1.48·11-s − 0.680·12-s − 0.237·13-s − 0.273·14-s + 2.16·15-s + 0.250·16-s − 0.168·17-s + 0.603·18-s + 0.229·19-s − 0.794·20-s + 0.526·21-s + 1.04·22-s + 0.317·23-s − 0.481·24-s + 1.52·25-s − 0.168·26-s + 0.200·27-s − 0.193·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9933822830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9933822830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 + 0.857T + 13T^{2} \) |
| 17 | \( 1 + 0.693T + 17T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 5.40T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 + 4.79T + 37T^{2} \) |
| 41 | \( 1 + 8.05T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 - 3.61T + 61T^{2} \) |
| 67 | \( 1 - 0.961T + 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 5.51T + 79T^{2} \) |
| 83 | \( 1 - 4.09T + 83T^{2} \) |
| 89 | \( 1 + 0.371T + 89T^{2} \) |
| 97 | \( 1 + 6.01T + 97T^{2} \) |
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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
−7.50123857690381252931401519861, −6.95551679466935881256178425154, −6.41888107218210443564175795274, −5.80376569163529886542035913245, −4.83909542719837630784222343057, −4.45085928193252933788776228881, −3.66488606488330071965479804035, −3.10575277771810402870361203206, −1.55271184723052644878681314443, −0.49203554390852255737659750069,
0.49203554390852255737659750069, 1.55271184723052644878681314443, 3.10575277771810402870361203206, 3.66488606488330071965479804035, 4.45085928193252933788776228881, 4.83909542719837630784222343057, 5.80376569163529886542035913245, 6.41888107218210443564175795274, 6.95551679466935881256178425154, 7.50123857690381252931401519861
