L(s) = 1 | − 1.19·2-s + 3.16·3-s − 0.576·4-s + 1.08·5-s − 3.77·6-s − 1.30·7-s + 3.07·8-s + 7.00·9-s − 1.29·10-s − 11-s − 1.82·12-s + 2.53·13-s + 1.56·14-s + 3.42·15-s − 2.51·16-s − 5.43·17-s − 8.35·18-s + 19-s − 0.623·20-s − 4.14·21-s + 1.19·22-s + 3.87·23-s + 9.72·24-s − 3.82·25-s − 3.03·26-s + 12.6·27-s + 0.754·28-s + ⋯ |
L(s) = 1 | − 0.843·2-s + 1.82·3-s − 0.288·4-s + 0.484·5-s − 1.54·6-s − 0.494·7-s + 1.08·8-s + 2.33·9-s − 0.408·10-s − 0.301·11-s − 0.526·12-s + 0.704·13-s + 0.417·14-s + 0.883·15-s − 0.628·16-s − 1.31·17-s − 1.96·18-s + 0.229·19-s − 0.139·20-s − 0.903·21-s + 0.254·22-s + 0.807·23-s + 1.98·24-s − 0.765·25-s − 0.594·26-s + 2.43·27-s + 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.305047396\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.305047396\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 - 3.16T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 + 6.11T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 0.992T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 - 5.84T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 7.01T + 83T^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−12.99309940489819925338778495044, −10.98995466386863249792893386981, −9.829265547862646932767143661423, −9.378878982306308230898716684408, −8.540992584434319441278181209512, −7.83105423894568512020074972486, −6.63082079478626454569283244052, −4.58390726684160789428592755383, −3.27911967032119883950331400134, −1.83487682027436614199988471346,
1.83487682027436614199988471346, 3.27911967032119883950331400134, 4.58390726684160789428592755383, 6.63082079478626454569283244052, 7.83105423894568512020074972486, 8.540992584434319441278181209512, 9.378878982306308230898716684408, 9.829265547862646932767143661423, 10.98995466386863249792893386981, 12.99309940489819925338778495044