L(s) = 1 | + (1.98 − 0.531i)2-s + (1.57 − 0.909i)3-s + (1.91 − 1.10i)4-s + (−0.737 − 2.75i)5-s + (2.64 − 2.64i)6-s + (0.311 − 0.311i)8-s + (0.155 − 0.269i)9-s + (−2.92 − 5.06i)10-s + (−0.616 − 0.165i)11-s + (2.01 − 3.48i)12-s + (3.40 − 1.19i)13-s + (−3.66 − 3.66i)15-s + (−1.76 + 3.05i)16-s + (2.16 + 3.74i)17-s + (0.165 − 0.616i)18-s + (−1.24 − 4.64i)19-s + ⋯ |
L(s) = 1 | + (1.40 − 0.375i)2-s + (0.909 − 0.525i)3-s + (0.958 − 0.553i)4-s + (−0.329 − 1.23i)5-s + (1.07 − 1.07i)6-s + (0.109 − 0.109i)8-s + (0.0518 − 0.0898i)9-s + (−0.924 − 1.60i)10-s + (−0.186 − 0.0498i)11-s + (0.581 − 1.00i)12-s + (0.943 − 0.330i)13-s + (−0.946 − 0.946i)15-s + (−0.440 + 0.763i)16-s + (0.524 + 0.908i)17-s + (0.0389 − 0.145i)18-s + (−0.285 − 1.06i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.78526 - 2.50319i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.78526 - 2.50319i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.40 + 1.19i)T \) |
good | 2 | \( 1 + (-1.98 + 0.531i)T + (1.73 - i)T^{2} \) |
| 3 | \( 1 + (-1.57 + 0.909i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.737 + 2.75i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (0.616 + 0.165i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.24 + 4.64i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.808 - 0.466i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 + (7.48 + 2.00i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-0.783 - 2.92i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.81 + 1.81i)T - 41iT^{2} \) |
| 43 | \( 1 - 10.4iT - 43T^{2} \) |
| 47 | \( 1 + (8.07 - 2.16i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.68 + 2.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.0935 - 0.349i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-6.74 - 3.89i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.66 + 9.94i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-5.56 - 5.56i)T + 71iT^{2} \) |
| 73 | \( 1 + (3.24 - 12.1i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.89 + 11.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.30 + 4.30i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.66 - 2.05i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (0.236 - 0.236i)T - 97iT^{2} \) |
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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
−10.75337241106349470905431240421, −9.267665037597403719319917332785, −8.435362008746821782828862982629, −8.017585594679051511374956380931, −6.58344877366112273325772604889, −5.50611304870835348184143102557, −4.69632602428741912070087026928, −3.71292346060938364653093651539, −2.76308784051988641525330237527, −1.44604788865539915507623225878,
2.56955860812261532399028333998, 3.48718879635302663482223316251, 3.88054614515744371744322112704, 5.19938868049235101841491947008, 6.26894821877490601895686888400, 6.98447447260026637492334660493, 7.972968949884646476122681970989, 9.024265998128497484635813915017, 9.978177547914400315018461749941, 10.88537313175216411594988528130