| 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
−12.80295512786879289147005725456, −11.45601071561937485733157413058, −10.19900519247517957623953205183, −9.316941846443093246981929019560, −8.664314225471307199778621311414, −7.41926729312402901315580685867, −6.35870850410031653974218227790, −5.23537426174211583547566522230, −3.14962103332485300174837737578, −2.03505256127745260128384146525,
1.48886973188113765631894691008, 3.78292035608529465112455474941, 4.83004980289436527500226770485, 5.69820056097191767995003698301, 7.78143519103849151882722020307, 8.694330544197241610549904518229, 9.313135089979760380195274316268, 10.17904350749282716679450238452, 10.87125134621904954092933076551, 12.94755268901818145893403944434
