| L(s) = 1 | + (−0.412 − 0.0196i)2-s + (1.15 + 1.28i)3-s + (−1.82 − 0.173i)4-s + (2.81 − 2.68i)5-s + (−0.452 − 0.553i)6-s + (0.539 − 2.79i)7-s + (1.56 + 0.224i)8-s + (−0.315 + 2.98i)9-s + (−1.21 + 1.05i)10-s + (−1.68 − 0.869i)11-s + (−1.88 − 2.54i)12-s + (2.45 − 0.473i)13-s + (−0.277 + 1.14i)14-s + (6.72 + 0.515i)15-s + (2.95 + 0.569i)16-s + (−2.19 + 4.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.291 − 0.0138i)2-s + (0.668 + 0.743i)3-s + (−0.910 − 0.0869i)4-s + (1.26 − 1.20i)5-s + (−0.184 − 0.225i)6-s + (0.203 − 1.05i)7-s + (0.553 + 0.0795i)8-s + (−0.105 + 0.994i)9-s + (−0.383 + 0.332i)10-s + (−0.508 − 0.262i)11-s + (−0.544 − 0.735i)12-s + (0.681 − 0.131i)13-s + (−0.0741 + 0.305i)14-s + (1.73 + 0.133i)15-s + (0.738 + 0.142i)16-s + (−0.533 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27159 - 0.163213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27159 - 0.163213i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.15 - 1.28i)T \) |
| 23 | \( 1 + (-1.47 - 4.56i)T \) |
| good | 2 | \( 1 + (0.412 + 0.0196i)T + (1.99 + 0.190i)T^{2} \) |
| 5 | \( 1 + (-2.81 + 2.68i)T + (0.237 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.539 + 2.79i)T + (-6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (1.68 + 0.869i)T + (6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-2.45 + 0.473i)T + (12.0 - 4.83i)T^{2} \) |
| 17 | \( 1 + (2.19 - 4.81i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.26 + 1.03i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.293 - 3.07i)T + (-28.4 + 5.48i)T^{2} \) |
| 31 | \( 1 + (7.06 + 5.55i)T + (7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.48 - 5.06i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (3.86 + 4.05i)T + (-1.95 + 40.9i)T^{2} \) |
| 43 | \( 1 + (0.990 + 1.25i)T + (-10.1 + 41.7i)T^{2} \) |
| 47 | \( 1 + (3.02 - 1.74i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.68 - 5.40i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (0.385 + 2.00i)T + (-54.7 + 21.9i)T^{2} \) |
| 61 | \( 1 + (-5.11 - 12.7i)T + (-44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (4.41 + 8.55i)T + (-38.8 + 54.5i)T^{2} \) |
| 71 | \( 1 + (0.382 - 0.595i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (1.51 + 3.31i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (8.46 + 2.92i)T + (62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (-9.59 - 9.15i)T + (3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (-2.44 - 16.9i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-17.4 + 4.23i)T + (86.2 - 44.4i)T^{2} \) |
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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
−12.94755268901818145893403944434, −10.87125134621904954092933076551, −10.17904350749282716679450238452, −9.313135089979760380195274316268, −8.694330544197241610549904518229, −7.78143519103849151882722020307, −5.69820056097191767995003698301, −4.83004980289436527500226770485, −3.78292035608529465112455474941, −1.48886973188113765631894691008,
2.03505256127745260128384146525, 3.14962103332485300174837737578, 5.23537426174211583547566522230, 6.35870850410031653974218227790, 7.41926729312402901315580685867, 8.664314225471307199778621311414, 9.316941846443093246981929019560, 10.19900519247517957623953205183, 11.45601071561937485733157413058, 12.80295512786879289147005725456
