| L(s) = 1 | + 4·2-s + 16·4-s + 158·7-s + 64·8-s + 148·11-s + 684·13-s + 632·14-s + 256·16-s − 2.04e3·17-s + 2.22e3·19-s + 592·22-s + 1.24e3·23-s + 2.73e3·26-s + 2.52e3·28-s + 270·29-s − 2.04e3·31-s + 1.02e3·32-s − 8.19e3·34-s − 4.37e3·37-s + 8.88e3·38-s + 2.39e3·41-s + 2.29e3·43-s + 2.36e3·44-s + 4.98e3·46-s + 1.06e4·47-s + 8.15e3·49-s + 1.09e4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.21·7-s + 0.353·8-s + 0.368·11-s + 1.12·13-s + 0.861·14-s + 1/4·16-s − 1.71·17-s + 1.41·19-s + 0.260·22-s + 0.491·23-s + 0.793·26-s + 0.609·28-s + 0.0596·29-s − 0.382·31-s + 0.176·32-s − 1.21·34-s − 0.525·37-s + 0.997·38-s + 0.222·41-s + 0.189·43-s + 0.184·44-s + 0.347·46-s + 0.705·47-s + 0.485·49-s + 0.561·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.493844114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.493844114\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 158 T + p^{5} T^{2} \) |
| 11 | \( 1 - 148 T + p^{5} T^{2} \) |
| 13 | \( 1 - 684 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2048 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2220 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1246 T + p^{5} T^{2} \) |
| 29 | \( 1 - 270 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2048 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4372 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2398 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2294 T + p^{5} T^{2} \) |
| 47 | \( 1 - 10682 T + p^{5} T^{2} \) |
| 53 | \( 1 + 2964 T + p^{5} T^{2} \) |
| 59 | \( 1 - 39740 T + p^{5} T^{2} \) |
| 61 | \( 1 + 42298 T + p^{5} T^{2} \) |
| 67 | \( 1 - 32098 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4248 T + p^{5} T^{2} \) |
| 73 | \( 1 - 30104 T + p^{5} T^{2} \) |
| 79 | \( 1 - 35280 T + p^{5} T^{2} \) |
| 83 | \( 1 - 27826 T + p^{5} T^{2} \) |
| 89 | \( 1 - 85210 T + p^{5} T^{2} \) |
| 97 | \( 1 + 97232 T + p^{5} T^{2} \) |
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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
−10.71007479777161546234847391137, −9.253632470481186327473785038370, −8.446933560814285810781459341124, −7.41948859321297140477752871317, −6.47227892820301167125022743763, −5.38725255058274861664687916643, −4.53563342875260244280185381903, −3.53904477299271211314632738635, −2.13072730422847242388823452591, −1.06742408108398779724254303666,
1.06742408108398779724254303666, 2.13072730422847242388823452591, 3.53904477299271211314632738635, 4.53563342875260244280185381903, 5.38725255058274861664687916643, 6.47227892820301167125022743763, 7.41948859321297140477752871317, 8.446933560814285810781459341124, 9.253632470481186327473785038370, 10.71007479777161546234847391137
