| L(s) = 1 | − 0.634i·2-s + 1.59·4-s + (−1.45 − 1.69i)5-s − 2.74i·7-s − 2.28i·8-s + (−1.07 + 0.924i)10-s − 4.00·11-s − i·13-s − 1.74·14-s + 1.74·16-s + 2.35i·17-s − 4.69·19-s + (−2.32 − 2.70i)20-s + 2.54i·22-s − 1.88i·23-s + ⋯ |
| L(s) = 1 | − 0.448i·2-s + 0.798·4-s + (−0.652 − 0.758i)5-s − 1.03i·7-s − 0.806i·8-s + (−0.340 + 0.292i)10-s − 1.20·11-s − 0.277i·13-s − 0.465·14-s + 0.437·16-s + 0.572i·17-s − 1.07·19-s + (−0.520 − 0.605i)20-s + 0.541i·22-s − 0.393i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.438006 - 1.18101i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.438006 - 1.18101i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.45 + 1.69i)T \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + 0.634iT - 2T^{2} \) |
| 7 | \( 1 + 2.74iT - 7T^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 17 | \( 1 - 2.35iT - 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 1.88iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 6.44iT - 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.691iT - 43T^{2} \) |
| 47 | \( 1 + 5.89iT - 47T^{2} \) |
| 53 | \( 1 + 9.14iT - 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 + 4.44T + 61T^{2} \) |
| 67 | \( 1 + 4.80iT - 67T^{2} \) |
| 71 | \( 1 - 1.82T + 71T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 8.42iT - 83T^{2} \) |
| 89 | \( 1 - 5.75T + 89T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−10.73161203174019161797166654166, −9.746630425870981082729717812619, −8.405297342091321298461850471888, −7.74680303233264800811654412971, −6.93165142773440694713922012093, −5.74769094694097728110114125532, −4.48130556845990488304212654890, −3.61795182270042896105339190723, −2.25280338604563465614128521563, −0.66701581722334161858827862976,
2.33412025650085740503457298136, 2.98778379512224121316770747987, 4.64268559810894432170619755816, 5.81553929989416171783510900325, 6.50726324696665747460729853307, 7.56918018842840230311742252813, 8.067420475217635924830514883074, 9.154614390423478338362551401151, 10.44504079967662405940520302902, 10.99618001106397908698080803204
