| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s − 2·11-s + 13-s + 16-s − 3·17-s + 4·19-s + 2·20-s − 2·22-s + 3·23-s − 25-s + 26-s + 7·29-s + 11·31-s + 32-s − 3·34-s − 6·37-s + 4·38-s + 2·40-s + 6·41-s + 43-s − 2·44-s + 3·46-s + 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.603·11-s + 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.917·19-s + 0.447·20-s − 0.426·22-s + 0.625·23-s − 1/5·25-s + 0.196·26-s + 1.29·29-s + 1.97·31-s + 0.176·32-s − 0.514·34-s − 0.986·37-s + 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.152·43-s − 0.301·44-s + 0.442·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.509315152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.509315152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.858491582708342556427559261956, −8.064645491147101931790045387521, −7.13790621913547938209696958174, −6.41371642316385572236262321411, −5.71712844391571482840343232501, −4.99614562130570219516186374288, −4.21966667354524075129226149758, −3.00004241953254677623600625591, −2.38283992470961307825516552466, −1.13028052123358941518074717677,
1.13028052123358941518074717677, 2.38283992470961307825516552466, 3.00004241953254677623600625591, 4.21966667354524075129226149758, 4.99614562130570219516186374288, 5.71712844391571482840343232501, 6.41371642316385572236262321411, 7.13790621913547938209696958174, 8.064645491147101931790045387521, 8.858491582708342556427559261956
