| L(s) = 1 | + (−0.328 + 1.37i)2-s + (−1.78 − 0.903i)4-s + 0.512i·5-s − 7-s + (1.82 − 2.15i)8-s + (−0.704 − 0.168i)10-s − 1.82i·11-s + 1.80i·13-s + (0.328 − 1.37i)14-s + (2.36 + 3.22i)16-s − 8.11·17-s + 3.43i·19-s + (0.462 − 0.914i)20-s + (2.51 + 0.600i)22-s + 3.65·23-s + ⋯ |
| L(s) = 1 | + (−0.232 + 0.972i)2-s + (−0.892 − 0.451i)4-s + 0.229i·5-s − 0.377·7-s + (0.646 − 0.763i)8-s + (−0.222 − 0.0531i)10-s − 0.551i·11-s + 0.500i·13-s + (0.0877 − 0.367i)14-s + (0.592 + 0.805i)16-s − 1.96·17-s + 0.789i·19-s + (0.103 − 0.204i)20-s + (0.536 + 0.128i)22-s + 0.762·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6888939009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6888939009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.328 - 1.37i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 0.512iT - 5T^{2} \) |
| 11 | \( 1 + 1.82iT - 11T^{2} \) |
| 13 | \( 1 - 1.80iT - 13T^{2} \) |
| 17 | \( 1 + 8.11T + 17T^{2} \) |
| 19 | \( 1 - 3.43iT - 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 7.98iT - 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + 8.44iT - 37T^{2} \) |
| 41 | \( 1 - 2.30T + 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 8.87iT - 53T^{2} \) |
| 59 | \( 1 + 2.50iT - 59T^{2} \) |
| 61 | \( 1 + 5.31iT - 61T^{2} \) |
| 67 | \( 1 - 6.44iT - 67T^{2} \) |
| 71 | \( 1 + 9.10T + 71T^{2} \) |
| 73 | \( 1 - 9.24T + 73T^{2} \) |
| 79 | \( 1 - 3.64T + 79T^{2} \) |
| 83 | \( 1 + 4.53iT - 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−9.094053637149990388388762743003, −8.679979082484824108763802960790, −7.72296499577278367343785981778, −6.78700834835321271618130263669, −6.39422697787984324268910567939, −5.41483817284346683507694412910, −4.44626769816258231518640109585, −3.58669075254953044981876899681, −2.09926923845619842078884369969, −0.32148682811248410987484957936,
1.22405786708977729674115013901, 2.52261667721422615268508103921, 3.28589403233717975008145396035, 4.63598664652102427374356504748, 4.91133290163392672543400020416, 6.43800833960977913989144084545, 7.19164483087349784313015504178, 8.269887655790039438850392516420, 9.033468489879030704440685313923, 9.428211252417786864003938892135
