L(s) = 1 | − 0.371·2-s + (−0.305 + 1.70i)3-s − 1.86·4-s + i·5-s + (0.113 − 0.634i)6-s + i·7-s + 1.43·8-s + (−2.81 − 1.04i)9-s − 0.371i·10-s + (1.93 + 2.69i)11-s + (0.569 − 3.17i)12-s + 6.33i·13-s − 0.371i·14-s + (−1.70 − 0.305i)15-s + 3.18·16-s − 1.53·17-s + ⋯ |
L(s) = 1 | − 0.262·2-s + (−0.176 + 0.984i)3-s − 0.930·4-s + 0.447i·5-s + (0.0464 − 0.258i)6-s + 0.377i·7-s + 0.507·8-s + (−0.937 − 0.347i)9-s − 0.117i·10-s + (0.583 + 0.811i)11-s + (0.164 − 0.916i)12-s + 1.75i·13-s − 0.0993i·14-s + (−0.440 − 0.0789i)15-s + 0.797·16-s − 0.373·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6368188562\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6368188562\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.305 - 1.70i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-1.93 - 2.69i)T \) |
good | 2 | \( 1 + 0.371T + 2T^{2} \) |
| 13 | \( 1 - 6.33iT - 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 1.59iT - 19T^{2} \) |
| 23 | \( 1 - 0.203iT - 23T^{2} \) |
| 29 | \( 1 - 2.40T + 29T^{2} \) |
| 31 | \( 1 + 5.91T + 31T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 + 9.29T + 41T^{2} \) |
| 43 | \( 1 - 9.25iT - 43T^{2} \) |
| 47 | \( 1 + 4.56iT - 47T^{2} \) |
| 53 | \( 1 - 2.51iT - 53T^{2} \) |
| 59 | \( 1 + 4.95iT - 59T^{2} \) |
| 61 | \( 1 + 8.69iT - 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 + 13.9iT - 71T^{2} \) |
| 73 | \( 1 + 7.53iT - 73T^{2} \) |
| 79 | \( 1 - 6.74iT - 79T^{2} \) |
| 83 | \( 1 + 5.23T + 83T^{2} \) |
| 89 | \( 1 + 10.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.65T + 97T^{2} \) |
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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
−9.970726026364870240844900597031, −9.430782583281481507108884534012, −8.989315931449803671516068522759, −8.042105454343546140357633819332, −6.85821991608430021578965006799, −6.03742152409132110050436268590, −4.82553312587331708213446619834, −4.33647903027873896634213854576, −3.42893599188486521950773125605, −1.87111116200563631591649177483,
0.36215754128752573331343067612, 1.22885042426074695084266069964, 2.93115745147043760400496546569, 4.04426858764199935320007512205, 5.26309995245388313207722687020, 5.80854060460704125422162777900, 6.97007979232507469018418652779, 7.86851460162197924786502245018, 8.455493591438469318083984935883, 9.063195345940924893090988079174