| L(s) = 1 | − 1.21i·2-s + 0.525·4-s + (2.21 + 0.311i)5-s − 2.90i·7-s − 3.06i·8-s + (0.377 − 2.68i)10-s − 0.214·11-s − i·13-s − 3.52·14-s − 2.67·16-s + 6.42i·17-s − 2.21·19-s + (1.16 + 0.163i)20-s + 0.260i·22-s − 4.68i·23-s + ⋯ |
| L(s) = 1 | − 0.858i·2-s + 0.262·4-s + (0.990 + 0.139i)5-s − 1.09i·7-s − 1.08i·8-s + (0.119 − 0.850i)10-s − 0.0646·11-s − 0.277i·13-s − 0.942·14-s − 0.668·16-s + 1.55i·17-s − 0.507·19-s + (0.260 + 0.0365i)20-s + 0.0554i·22-s − 0.977i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27461 - 1.46621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27461 - 1.46621i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.21 - 0.311i)T \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 + 1.21iT - 2T^{2} \) |
| 7 | \( 1 + 2.90iT - 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 17 | \( 1 - 6.42iT - 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 + 4.68iT - 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 + 5.59T + 31T^{2} \) |
| 37 | \( 1 + 2.28iT - 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 - 6.36iT - 43T^{2} \) |
| 47 | \( 1 - 1.09iT - 47T^{2} \) |
| 53 | \( 1 - 6.23iT - 53T^{2} \) |
| 59 | \( 1 + 9.26T + 59T^{2} \) |
| 61 | \( 1 + 0.280T + 61T^{2} \) |
| 67 | \( 1 - 7.76iT - 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 + 10.2iT - 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 9.52iT - 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 - 18.0iT - 97T^{2} \) |
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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
−10.57807684692451668613535050356, −10.04491122399045618520275176755, −8.944868445946318500865310763778, −7.79049773070996641021197458474, −6.66449881261219603530823098679, −6.12054843855269643828899972711, −4.58219796518417560285471589218, −3.52238445067189867829646812452, −2.33820813853914405151011278517, −1.19725618245584263065815061063,
1.95419050851494371909103058201, 2.88363842836354740083778552999, 4.96214973605473106653766755151, 5.51557856025886068045755947593, 6.42448031087024221925532532203, 7.14967092065387098046028798242, 8.336285910036338377717276576716, 9.055110410939844463147126560529, 9.834020409792698250882933683089, 10.97972765218085866983429156796
