L(s) = 1 | + (1.11 − 1.32i)3-s + (1.43 + 2.49i)5-s + (−0.5 + 0.866i)7-s + (−0.520 − 2.95i)9-s + (0.592 − 1.02i)11-s + (2.37 + 4.12i)13-s + (4.91 + 0.866i)15-s + 5.41·17-s + 1.10·19-s + (0.592 + 1.62i)21-s + (−2.95 − 5.11i)23-s + (−1.64 + 2.84i)25-s + (−4.5 − 2.59i)27-s + (−2.49 + 4.31i)29-s + (2.78 + 4.82i)31-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)3-s + (0.643 + 1.11i)5-s + (−0.188 + 0.327i)7-s + (−0.173 − 0.984i)9-s + (0.178 − 0.309i)11-s + (0.659 + 1.14i)13-s + (1.26 + 0.223i)15-s + 1.31·17-s + 0.253·19-s + (0.129 + 0.355i)21-s + (−0.615 − 1.06i)23-s + (−0.329 + 0.569i)25-s + (−0.866 − 0.499i)27-s + (−0.462 + 0.801i)29-s + (0.500 + 0.866i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.97481\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97481\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.11 + 1.32i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.43 - 2.49i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.592 + 1.02i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.37 - 4.12i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 - 1.10T + 19T^{2} \) |
| 23 | \( 1 + (2.95 + 5.11i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.49 - 4.31i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.78 - 4.82i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.42T + 37T^{2} \) |
| 41 | \( 1 + (3.81 + 6.61i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.78 + 6.55i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.141 - 0.245i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4.22T + 53T^{2} \) |
| 59 | \( 1 + (5.86 + 10.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.992 - 1.71i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.76 - 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.11T + 71T^{2} \) |
| 73 | \( 1 + 0.327T + 73T^{2} \) |
| 79 | \( 1 + (-5.24 + 9.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.62 - 6.28i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + (9.04 - 15.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.86678809587354435708560551334, −9.983341486684717662991122119401, −9.070469338324904231909199166086, −8.276925895457336461886466769818, −7.06227836801114093530091969354, −6.53400361979604955585751058407, −5.63908172909303715452789852848, −3.73938250427486782355641103056, −2.81590973226911545333304764240, −1.65958062063998843489816915992,
1.40121906865435043672717148394, 3.08313116286369443700034878597, 4.13781827320797941547118795192, 5.25426861693353173958739864670, 5.93428098205416709517535370126, 7.73867087884380480464311255400, 8.205430091053355367752803736005, 9.486653734529703568119013503777, 9.689966219888660801765952797025, 10.64801881568062147750559432836