| L(s) = 1 | − 2.21·2-s + 2.88·4-s + 1.31·7-s − 1.96·8-s + 4.29·11-s + 2.97·13-s − 2.91·14-s − 1.43·16-s + 0.642·17-s − 5.07·19-s − 9.48·22-s + 8.84·23-s − 6.57·26-s + 3.80·28-s − 29-s − 6.27·31-s + 7.10·32-s − 1.42·34-s − 0.934·37-s + 11.2·38-s + 11.0·41-s − 2.03·43-s + 12.3·44-s − 19.5·46-s + 9.57·47-s − 5.26·49-s + 8.59·52-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.44·4-s + 0.497·7-s − 0.693·8-s + 1.29·11-s + 0.825·13-s − 0.777·14-s − 0.359·16-s + 0.155·17-s − 1.16·19-s − 2.02·22-s + 1.84·23-s − 1.29·26-s + 0.718·28-s − 0.185·29-s − 1.12·31-s + 1.25·32-s − 0.243·34-s − 0.153·37-s + 1.82·38-s + 1.73·41-s − 0.311·43-s + 1.86·44-s − 2.88·46-s + 1.39·47-s − 0.752·49-s + 1.19·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.149480921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.149480921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 - 0.642T + 17T^{2} \) |
| 19 | \( 1 + 5.07T + 19T^{2} \) |
| 23 | \( 1 - 8.84T + 23T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 + 0.934T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 - 9.57T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 + 2.55T + 59T^{2} \) |
| 61 | \( 1 - 8.46T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 5.10T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 3.30T + 79T^{2} \) |
| 83 | \( 1 - 5.24T + 83T^{2} \) |
| 89 | \( 1 + 5.05T + 89T^{2} \) |
| 97 | \( 1 - 2.04T + 97T^{2} \) |
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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
−8.159426086225580764521116838056, −7.50063298534563337183898801465, −6.73250924967370053620396349675, −6.32361593390043195202639683875, −5.24515344907496627410145832019, −4.30685404334794276469933974745, −3.53334614313887340554896238582, −2.32016921242010620489014593874, −1.48610245672610154005068073947, −0.77582523477395432958171402416,
0.77582523477395432958171402416, 1.48610245672610154005068073947, 2.32016921242010620489014593874, 3.53334614313887340554896238582, 4.30685404334794276469933974745, 5.24515344907496627410145832019, 6.32361593390043195202639683875, 6.73250924967370053620396349675, 7.50063298534563337183898801465, 8.159426086225580764521116838056
