L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.5i)3-s + (0.866 − 0.499i)4-s + (−1.11 − 4.17i)5-s + (−0.707 + 0.707i)6-s + (2.62 + 0.344i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (2.16 + 3.74i)10-s + (−4.88 − 1.30i)11-s + (0.5 − 0.866i)12-s + (3.28 − 1.48i)13-s + (−2.62 + 0.346i)14-s + (−3.05 − 3.05i)15-s + (0.500 − 0.866i)16-s + (1.53 + 2.65i)17-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (−0.500 − 1.86i)5-s + (−0.288 + 0.288i)6-s + (0.991 + 0.130i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.683 + 1.18i)10-s + (−1.47 − 0.394i)11-s + (0.144 − 0.249i)12-s + (0.911 − 0.410i)13-s + (−0.701 + 0.0925i)14-s + (−0.789 − 0.789i)15-s + (0.125 − 0.216i)16-s + (0.372 + 0.644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.602657 - 0.898938i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.602657 - 0.898938i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.62 - 0.344i)T \) |
| 13 | \( 1 + (-3.28 + 1.48i)T \) |
good | 5 | \( 1 + (1.11 + 4.17i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (4.88 + 1.30i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.53 - 2.65i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.919 + 3.43i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.27 - 1.89i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.32T + 29T^{2} \) |
| 31 | \( 1 + (7.76 + 2.08i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.00 - 3.74i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.16 + 2.16i)T - 41iT^{2} \) |
| 43 | \( 1 + 8.74iT - 43T^{2} \) |
| 47 | \( 1 + (-8.63 + 2.31i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.196 + 0.339i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.217 + 0.810i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.31 - 1.91i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.57 + 5.85i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.51 - 1.51i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.638 - 2.38i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.16 + 10.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.36 + 6.36i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.53 + 1.74i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (9.94 - 9.94i)T - 97iT^{2} \) |
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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
−10.56741988591302228368398254920, −9.100775965032224551457521922363, −8.753993923225454540167037845199, −7.943346537924152031225194962111, −7.55582505831923438798899069280, −5.65038985269625778419221693249, −5.11950902276319772362908312368, −3.76353817586839743917011802544, −1.96885856455891871105317461241, −0.74422704287401577811641726000,
2.09809111078169665181279910978, 3.05140139983382028022966070849, 4.08511172433462391471354042552, 5.63532136197051495017919522541, 7.01244728031990201795519368046, 7.62405190708336625904828476472, 8.199387047646474371132429663482, 9.435410478115131373061082686550, 10.42865360499895402011354609178, 10.93320010845022121481418378031