L(s) = 1 | + (−0.258 − 0.965i)2-s + (−1.69 − 0.378i)3-s + (−0.866 + 0.499i)4-s + (0.593 + 2.15i)5-s + (0.0717 + 1.73i)6-s + (2.64 + 0.113i)7-s + (0.707 + 0.707i)8-s + (2.71 + 1.27i)9-s + (1.92 − 1.13i)10-s + (−3.54 + 2.04i)11-s + (1.65 − 0.517i)12-s + (3.69 − 3.69i)13-s + (−0.574 − 2.58i)14-s + (−0.186 − 3.86i)15-s + (0.500 − 0.866i)16-s + (7.20 + 1.93i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.975 − 0.218i)3-s + (−0.433 + 0.249i)4-s + (0.265 + 0.964i)5-s + (0.0292 + 0.706i)6-s + (0.999 + 0.0427i)7-s + (0.249 + 0.249i)8-s + (0.904 + 0.426i)9-s + (0.609 − 0.357i)10-s + (−1.06 + 0.616i)11-s + (0.477 − 0.149i)12-s + (1.02 − 1.02i)13-s + (−0.153 − 0.690i)14-s + (−0.0482 − 0.998i)15-s + (0.125 − 0.216i)16-s + (1.74 + 0.468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.907592 - 0.0741129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.907592 - 0.0741129i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (1.69 + 0.378i)T \) |
| 5 | \( 1 + (-0.593 - 2.15i)T \) |
| 7 | \( 1 + (-2.64 - 0.113i)T \) |
good | 11 | \( 1 + (3.54 - 2.04i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.69 + 3.69i)T - 13iT^{2} \) |
| 17 | \( 1 + (-7.20 - 1.93i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.12 - 1.22i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.06 - 0.284i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 + (-2.62 - 4.54i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.34 - 1.43i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 6.74iT - 41T^{2} \) |
| 43 | \( 1 + (-1.27 + 1.27i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.796 + 2.97i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.47 + 5.50i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (5.09 + 8.81i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.814 - 1.41i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.909 + 3.39i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 5.89iT - 71T^{2} \) |
| 73 | \( 1 + (5.38 + 1.44i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (8.18 + 4.72i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.31 + 6.31i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.71 + 11.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.21 + 5.21i)T + 97iT^{2} \) |
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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
−12.14218591310631467840000364280, −11.28045049779544932640460937385, −10.38217528432872353629859187599, −10.12509306471288350757460793104, −8.122013008190373107364458507788, −7.47067064706113196704840965066, −5.89778705330190936052500655602, −5.06309270433078581154587581712, −3.32140861130207095764731067434, −1.58237961927791764783206382900,
1.15487606928045393639366501820, 4.20171551550435632133974117913, 5.33363541393493799891991061284, 5.80432246414123165332357124577, 7.38583323852326003409987259180, 8.333759760275878039370501664656, 9.375579428220257811404098612982, 10.44033207520884433394263395924, 11.44602808282655534388822364296, 12.28113922119191019806765121195