L(s) = 1 | + 0.246·2-s − 1.93·4-s + (−2.15 − 0.608i)5-s + 2.59·7-s − 0.971·8-s + (−0.530 − 0.150i)10-s + 3.81i·11-s + (1.54 − 3.25i)13-s + 0.639·14-s + 3.63·16-s + 4.63i·17-s + 5.02i·19-s + (4.17 + 1.18i)20-s + 0.939i·22-s − 4i·23-s + ⋯ |
L(s) = 1 | + 0.174·2-s − 0.969·4-s + (−0.962 − 0.272i)5-s + 0.979·7-s − 0.343·8-s + (−0.167 − 0.0474i)10-s + 1.14i·11-s + (0.427 − 0.903i)13-s + 0.170·14-s + 0.909·16-s + 1.12i·17-s + 1.15i·19-s + (0.932 + 0.263i)20-s + 0.200i·22-s − 0.834i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.992838 + 0.451072i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.992838 + 0.451072i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.15 + 0.608i)T \) |
| 13 | \( 1 + (-1.54 + 3.25i)T \) |
good | 2 | \( 1 - 0.246T + 2T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 11 | \( 1 - 3.81iT - 11T^{2} \) |
| 17 | \( 1 - 4.63iT - 17T^{2} \) |
| 19 | \( 1 - 5.02iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 - 10.2iT - 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 - 6.40iT - 41T^{2} \) |
| 43 | \( 1 - 0.639iT - 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + 6.51iT - 53T^{2} \) |
| 59 | \( 1 - 6.24iT - 59T^{2} \) |
| 61 | \( 1 + 3.23T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 13.4iT - 71T^{2} \) |
| 73 | \( 1 - 3.08T + 73T^{2} \) |
| 79 | \( 1 + 3.36T + 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 + 8.83iT - 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−10.63748394906284137453020456985, −10.14337298603645839843094599530, −8.731881973275026652813777221451, −8.264556880581166321723363154614, −7.59300248692604949960868367334, −6.14615277468195951459007291726, −4.87670674628627103262690560428, −4.43705729034987961539627656432, −3.32886793342450437273359967647, −1.30437283698218140755700925701,
0.72227736739406260464617562700, 2.90609945819445959736187370985, 4.10627934392375713195881601266, 4.74183180545785364117460531167, 5.86061646148262865529717624977, 7.15444186953401186919775436416, 8.062849053808430888745088092609, 8.748758979168819271356666299910, 9.494983978033590000689710896111, 10.87722570715548020823318907635