L(s) = 1 | + (0.931 + 1.06i)2-s + (−0.268 − 0.527i)3-s + (−0.265 + 1.98i)4-s + (0.994 + 2.00i)5-s + (0.311 − 0.777i)6-s + (−0.707 − 0.707i)7-s + (−2.35 + 1.56i)8-s + (1.55 − 2.14i)9-s + (−1.20 + 2.92i)10-s + (1.31 + 1.81i)11-s + (1.11 − 0.392i)12-s + (0.761 + 4.80i)13-s + (0.0942 − 1.41i)14-s + (0.789 − 1.06i)15-s + (−3.85 − 1.05i)16-s + (3.70 + 1.88i)17-s + ⋯ |
L(s) = 1 | + (0.658 + 0.752i)2-s + (−0.155 − 0.304i)3-s + (−0.132 + 0.991i)4-s + (0.444 + 0.895i)5-s + (0.127 − 0.317i)6-s + (−0.267 − 0.267i)7-s + (−0.833 + 0.552i)8-s + (0.519 − 0.714i)9-s + (−0.381 + 0.924i)10-s + (0.397 + 0.546i)11-s + (0.322 − 0.113i)12-s + (0.211 + 1.33i)13-s + (0.0251 − 0.377i)14-s + (0.203 − 0.274i)15-s + (−0.964 − 0.263i)16-s + (0.897 + 0.457i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10235 + 1.74217i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10235 + 1.74217i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.931 - 1.06i)T \) |
| 5 | \( 1 + (-0.994 - 2.00i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (0.268 + 0.527i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-1.31 - 1.81i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.761 - 4.80i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-3.70 - 1.88i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.35 + 4.16i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.13 - 7.19i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (4.52 + 1.46i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (3.25 - 1.05i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.224 + 0.0355i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-5.11 - 3.71i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-7.56 + 7.56i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.175 - 0.0892i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-11.1 + 5.68i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (2.11 + 1.53i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (9.73 - 7.07i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.74 + 13.2i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (2.08 + 0.678i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.792 + 0.125i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-1.50 + 4.61i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-14.2 - 7.25i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-1.36 - 1.87i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.59 - 9.00i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.88400331554254355473617161226, −9.552580944590006085650655841084, −9.200044211317184526349811820772, −7.58017658230133434253229910623, −7.07033191579626209083430873990, −6.40362271721242299730846708032, −5.61207106627994385523106262962, −4.16234485940324520811805905407, −3.49594170221833656300214037133, −1.94284240057086774033346757006,
0.948899524964243198415916499924, 2.34625718760336043662065863091, 3.64959348976049959243823119399, 4.62243847098948275976439212964, 5.61477181971826436642925229312, 5.99207514812747363156745100708, 7.66053577502652378073720184584, 8.698653055477938274962700150486, 9.570884891500499565496175131374, 10.32721611006028649608433747183