L(s) = 1 | + (−0.258 − 0.965i)2-s + (1.72 − 0.158i)3-s + (−0.866 + 0.499i)4-s + (−1.67 + 1.48i)5-s + (−0.599 − 1.62i)6-s + (−2.12 − 2.12i)7-s + (0.707 + 0.707i)8-s + (2.94 − 0.548i)9-s + (1.86 + 1.23i)10-s − 5.82i·11-s + (−1.41 + 0.999i)12-s + (2.73 + 0.732i)13-s + (−1.5 + 2.59i)14-s + (−2.64 + 2.82i)15-s + (0.500 − 0.866i)16-s + (−1.61 − 6.02i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (0.995 − 0.0917i)3-s + (−0.433 + 0.249i)4-s + (−0.748 + 0.663i)5-s + (−0.244 − 0.663i)6-s + (−0.801 − 0.801i)7-s + (0.249 + 0.249i)8-s + (0.983 − 0.182i)9-s + (0.590 + 0.389i)10-s − 1.75i·11-s + (−0.408 + 0.288i)12-s + (0.757 + 0.203i)13-s + (−0.400 + 0.694i)14-s + (−0.684 + 0.729i)15-s + (0.125 − 0.216i)16-s + (−0.391 − 1.46i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.774658 - 1.12386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.774658 - 1.12386i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (-1.72 + 0.158i)T \) |
| 5 | \( 1 + (1.67 - 1.48i)T \) |
| 19 | \( 1 + (-4.33 - 0.5i)T \) |
good | 7 | \( 1 + (2.12 + 2.12i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.82iT - 11T^{2} \) |
| 13 | \( 1 + (-2.73 - 0.732i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (1.61 + 6.02i)T + (-14.7 + 8.5i)T^{2} \) |
| 23 | \( 1 + (0.776 - 2.89i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (4.41 + 7.64i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + (-2.12 - 2.12i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.09 - 4.09i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-10.1 - 2.71i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.93 - 7.23i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (2.82 - 4.89i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.12 - 5.40i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.46 - 5.46i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (11.6 + 6.70i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.00 + 7.49i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.08 - 5.24i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 - 3i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.08 + 5.34i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.40 - 0.643i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.50401308027178355219508178708, −9.542553051019438192637632762496, −8.867887153922733309867154327803, −7.80196985815174298294940766356, −7.23529209018696132545333500829, −6.03155528906033273756823180451, −4.16407841422667702518966751780, −3.42891902906938218913035773438, −2.82001137644116429215567951312, −0.77620951638236886601840778658,
1.80781441363620314061739509994, 3.46399984253765825144869972949, 4.34793364933986064805913305729, 5.46104647305219259711159643968, 6.77612826542098814828164064438, 7.53816050585118443209110023505, 8.449555499228544764510597320654, 9.064221309004104481735627040882, 9.721433388919155245427960213812, 10.74503091737723960220820286300