L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s − 3.73i·5-s + (−0.866 + 0.499i)6-s + (−2.36 + 1.36i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (1.86 − 3.23i)10-s + (−1.09 − 0.633i)11-s − 0.999·12-s − 2.73·14-s + (3.23 + 1.86i)15-s + (−0.5 + 0.866i)16-s + (−2.86 − 4.96i)17-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s − 1.66i·5-s + (−0.353 + 0.204i)6-s + (−0.894 + 0.516i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.590 − 1.02i)10-s + (−0.331 − 0.191i)11-s − 0.288·12-s − 0.730·14-s + (0.834 + 0.481i)15-s + (−0.125 + 0.216i)16-s + (−0.695 − 1.20i)17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8551107632\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8551107632\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3.73iT - 5T^{2} \) |
| 7 | \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.09 + 0.633i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.09 + 3.63i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.23 + 3.86i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.46iT - 31T^{2} \) |
| 37 | \( 1 + (3.06 + 1.76i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.13 + 4.69i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.83 + 8.36i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.19iT - 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.59 - 7.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 6.56i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.09 - 2.36i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6.26iT - 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 - 0.196iT - 83T^{2} \) |
| 89 | \( 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.566142673412678641061300229868, −8.791966465346618534900672602801, −8.329884144295394286809012506481, −6.95342061251411452913824849256, −6.05946124444720685329282293368, −5.23399235460784412676336116036, −4.65214218448244023985792314115, −3.68702409204954292201059605349, −2.34213998501703105263335673941, −0.30343879879023789325653856103,
1.90140886989601695175632634054, 3.00471344361325431322676938020, 3.67662596668744703938691514945, 4.96930059717601724539577696471, 6.39283012958816740439915082328, 6.52008279617424131326157016970, 7.27905551182654011068928738535, 8.423333808405596985681771643899, 9.826244437158319361769321732547, 10.40972739524467484745132970690