| L(s) = 1 | + (−2.15 + 0.603i)5-s − 0.703i·7-s + 3.60·11-s + 2.28i·13-s − 0.471i·17-s − 5.51·19-s + 3.10i·23-s + (4.27 − 2.59i)25-s − 0.391·29-s + 6.21·31-s + (0.424 + 1.51i)35-s − 10.1i·37-s − 10.3·41-s + 0.322i·43-s + 10.9i·47-s + ⋯ |
| L(s) = 1 | + (−0.962 + 0.269i)5-s − 0.265i·7-s + 1.08·11-s + 0.633i·13-s − 0.114i·17-s − 1.26·19-s + 0.647i·23-s + (0.854 − 0.519i)25-s − 0.0727·29-s + 1.11·31-s + (0.0716 + 0.255i)35-s − 1.66i·37-s − 1.61·41-s + 0.0491i·43-s + 1.60i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9750429417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9750429417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.15 - 0.603i)T \) |
| good | 7 | \( 1 + 0.703iT - 7T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 - 2.28iT - 13T^{2} \) |
| 17 | \( 1 + 0.471iT - 17T^{2} \) |
| 19 | \( 1 + 5.51T + 19T^{2} \) |
| 23 | \( 1 - 3.10iT - 23T^{2} \) |
| 29 | \( 1 + 0.391T + 29T^{2} \) |
| 31 | \( 1 - 6.21T + 31T^{2} \) |
| 37 | \( 1 + 10.1iT - 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.322iT - 43T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 - 12.0iT - 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 + 2.18T + 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 3.13T + 71T^{2} \) |
| 73 | \( 1 - 3.39iT - 73T^{2} \) |
| 79 | \( 1 - 3.33T + 79T^{2} \) |
| 83 | \( 1 - 14.1iT - 83T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 - 13.2iT - 97T^{2} \) |
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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
−8.918382226739795757652624213809, −8.065519212471631218500694346339, −7.38260617165325955656342244675, −6.65369074797612367938911151472, −6.10266333515006592317668322740, −4.79068487789979432110406464249, −4.12921421594834212954336529720, −3.55279744971601131644902784701, −2.38227428016324725073503897146, −1.13998382764174582593110442141,
0.33840586654153983767670696450, 1.62402849534834177128828599209, 2.90546810963772321321179743334, 3.77379389673313118841116258280, 4.48798742310735682799573014724, 5.25200612940957416554265267557, 6.42295476263464659009634402119, 6.78303204736066924244295707022, 7.85241264993848138664295883500, 8.597982697526827666220472247851
