| L(s) = 1 | − 2.18·3-s − 1.66·5-s + 1.81·7-s + 1.76·9-s + 3.97·11-s − 4.88·13-s + 3.63·15-s − 3.09·17-s + 3.71·19-s − 3.96·21-s − 2.52·23-s − 2.22·25-s + 2.70·27-s + 4.16·29-s − 4.17·31-s − 8.67·33-s − 3.02·35-s + 11.0·37-s + 10.6·39-s − 0.0987·41-s + 2.04·43-s − 2.93·45-s − 0.469·47-s − 3.69·49-s + 6.75·51-s − 3.85·53-s − 6.62·55-s + ⋯ |
| L(s) = 1 | − 1.25·3-s − 0.745·5-s + 0.686·7-s + 0.587·9-s + 1.19·11-s − 1.35·13-s + 0.939·15-s − 0.750·17-s + 0.853·19-s − 0.865·21-s − 0.525·23-s − 0.444·25-s + 0.519·27-s + 0.772·29-s − 0.750·31-s − 1.51·33-s − 0.512·35-s + 1.81·37-s + 1.70·39-s − 0.0154·41-s + 0.312·43-s − 0.437·45-s − 0.0684·47-s − 0.528·49-s + 0.946·51-s − 0.530·53-s − 0.893·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8133126253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8133126253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 - 3.97T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 3.71T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 0.0987T + 41T^{2} \) |
| 43 | \( 1 - 2.04T + 43T^{2} \) |
| 47 | \( 1 + 0.469T + 47T^{2} \) |
| 53 | \( 1 + 3.85T + 53T^{2} \) |
| 59 | \( 1 + 7.09T + 59T^{2} \) |
| 61 | \( 1 + 0.640T + 61T^{2} \) |
| 67 | \( 1 + 1.93T + 67T^{2} \) |
| 71 | \( 1 + 0.867T + 71T^{2} \) |
| 73 | \( 1 + 6.51T + 73T^{2} \) |
| 79 | \( 1 - 9.82T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 9.81T + 89T^{2} \) |
| 97 | \( 1 + 1.98T + 97T^{2} \) |
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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
−8.290859415889488888414749026149, −7.60205576513450234333308839614, −6.95754898396719854932526685134, −6.20068000089059005164426222289, −5.47760951103174953285297786067, −4.55110644535511700384123297887, −4.29614596372741741860012632818, −3.00179615210389152173555874428, −1.73244615216372368272700576123, −0.55647462105700252076438897002,
0.55647462105700252076438897002, 1.73244615216372368272700576123, 3.00179615210389152173555874428, 4.29614596372741741860012632818, 4.55110644535511700384123297887, 5.47760951103174953285297786067, 6.20068000089059005164426222289, 6.95754898396719854932526685134, 7.60205576513450234333308839614, 8.290859415889488888414749026149
