L(s) = 1 | + 0.231i·2-s + (0.707 + 0.707i)3-s + 1.94·4-s + 2.07i·5-s + (−0.163 + 0.163i)6-s + (−3.15 − 3.15i)7-s + 0.912i·8-s + 1.00i·9-s − 0.479·10-s + (1.22 + 1.22i)11-s + (1.37 + 1.37i)12-s + (−1.95 − 1.95i)13-s + (0.730 − 0.730i)14-s + (−1.46 + 1.46i)15-s + 3.68·16-s + (3.82 − 3.82i)17-s + ⋯ |
L(s) = 1 | + 0.163i·2-s + (0.408 + 0.408i)3-s + 0.973·4-s + 0.926i·5-s + (−0.0667 + 0.0667i)6-s + (−1.19 − 1.19i)7-s + 0.322i·8-s + 0.333i·9-s − 0.151·10-s + (0.368 + 0.368i)11-s + (0.397 + 0.397i)12-s + (−0.541 − 0.541i)13-s + (0.195 − 0.195i)14-s + (−0.378 + 0.378i)15-s + 0.920·16-s + (0.928 − 0.928i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22479 + 0.424927i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22479 + 0.424927i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (2.81 - 5.75i)T \) |
good | 2 | \( 1 - 0.231iT - 2T^{2} \) |
| 5 | \( 1 - 2.07iT - 5T^{2} \) |
| 7 | \( 1 + (3.15 + 3.15i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.22 - 1.22i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.95 + 1.95i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.82 + 3.82i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.943 - 0.943i)T - 19iT^{2} \) |
| 23 | \( 1 + 9.01T + 23T^{2} \) |
| 29 | \( 1 + (4.83 + 4.83i)T + 29iT^{2} \) |
| 31 | \( 1 + 0.770T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 43 | \( 1 + 6.89iT - 43T^{2} \) |
| 47 | \( 1 + (3.10 - 3.10i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.02 - 5.02i)T + 53iT^{2} \) |
| 59 | \( 1 + 1.57T + 59T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 + (-5.14 + 5.14i)T - 67iT^{2} \) |
| 71 | \( 1 + (-1.55 - 1.55i)T + 71iT^{2} \) |
| 73 | \( 1 - 11.1iT - 73T^{2} \) |
| 79 | \( 1 + (-3.45 - 3.45i)T + 79iT^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 89 | \( 1 + (7.29 + 7.29i)T + 89iT^{2} \) |
| 97 | \( 1 + (7.83 - 7.83i)T - 97iT^{2} \) |
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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
−13.80098525707577075232013040034, −12.48739529632815436032820654097, −11.32801459139569645243624728846, −10.07616749153555185746729634701, −9.937263824732925545261024569974, −7.74050313603463558471230960386, −7.09262092324848688245169781941, −6.00374952810281012358308697849, −3.83751378130709934930874444500, −2.73139305865347067315926200028,
2.01195665824630835541122870824, 3.50043032591481155034620909165, 5.70960932002848369672009528246, 6.57462226181034195776439537203, 8.023925243243617039643034026128, 9.092273828544042874417449486507, 10.00225277553450131671141364423, 11.64208172919724266990571072947, 12.43498684630531667151849576708, 12.84988683896094497371815248553