| L(s) = 1 | − 2.49·2-s + 4.24·4-s + 1.07·5-s − 0.193·7-s − 5.61·8-s − 2.67·10-s − 0.571·11-s − 5.68·13-s + 0.483·14-s + 5.54·16-s − 2.04·17-s + 4.54·20-s + 1.42·22-s + 1.55·23-s − 3.85·25-s + 14.2·26-s − 0.822·28-s − 6.09·29-s − 5.86·31-s − 2.62·32-s + 5.11·34-s − 0.207·35-s − 9.36·37-s − 6.01·40-s + 1.92·41-s − 3.21·43-s − 2.42·44-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.12·4-s + 0.478·5-s − 0.0731·7-s − 1.98·8-s − 0.846·10-s − 0.172·11-s − 1.57·13-s + 0.129·14-s + 1.38·16-s − 0.496·17-s + 1.01·20-s + 0.304·22-s + 0.324·23-s − 0.770·25-s + 2.78·26-s − 0.155·28-s − 1.13·29-s − 1.05·31-s − 0.463·32-s + 0.876·34-s − 0.0350·35-s − 1.53·37-s − 0.950·40-s + 0.301·41-s − 0.489·43-s − 0.366·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9747 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9747 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3839317187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3839317187\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 7 | \( 1 + 0.193T + 7T^{2} \) |
| 11 | \( 1 + 0.571T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 23 | \( 1 - 1.55T + 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 + 5.86T + 31T^{2} \) |
| 37 | \( 1 + 9.36T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 7.69T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 8.02T + 61T^{2} \) |
| 67 | \( 1 - 6.74T + 67T^{2} \) |
| 71 | \( 1 - 7.00T + 71T^{2} \) |
| 73 | \( 1 + 8.07T + 73T^{2} \) |
| 79 | \( 1 - 4.88T + 79T^{2} \) |
| 83 | \( 1 + 6.02T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 7.59T + 97T^{2} \) |
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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
−7.72706013780863591489809382121, −7.12856760525456083929871890567, −6.78657232941965838656472494917, −5.73894502813851500377750460279, −5.21251334407420818500896986392, −4.10583917995621105616389982261, −2.99179472791759598116196344897, −2.16654137930229823688513105428, −1.72036054104962100415894731557, −0.36861468132550081678006057678,
0.36861468132550081678006057678, 1.72036054104962100415894731557, 2.16654137930229823688513105428, 2.99179472791759598116196344897, 4.10583917995621105616389982261, 5.21251334407420818500896986392, 5.73894502813851500377750460279, 6.78657232941965838656472494917, 7.12856760525456083929871890567, 7.72706013780863591489809382121
