L(s) = 1 | + (−1 + 1.41i)3-s + (−1.73 + 1.41i)5-s + (1 − 2.44i)7-s + (−1.00 − 2.82i)9-s + 2.82i·11-s + 4·13-s + (−0.267 − 3.86i)15-s + 2.82i·17-s + (2.46 + 3.86i)21-s − 3.46·23-s + (0.999 − 4.89i)25-s + (5.00 + 1.41i)27-s + 5.65i·29-s + 9.79i·31-s + (−4.00 − 2.82i)33-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (−0.774 + 0.632i)5-s + (0.377 − 0.925i)7-s + (−0.333 − 0.942i)9-s + 0.852i·11-s + 1.10·13-s + (−0.0691 − 0.997i)15-s + 0.685i·17-s + (0.537 + 0.843i)21-s − 0.722·23-s + (0.199 − 0.979i)25-s + (0.962 + 0.272i)27-s + 1.05i·29-s + 1.75i·31-s + (−0.696 − 0.492i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7150500850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7150500850\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 5 | \( 1 + (1.73 - 1.41i)T \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 9.79iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 4.89iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 9.79iT - 61T^{2} \) |
| 67 | \( 1 - 4.89iT - 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.05791294510337668294922814590, −8.851732665644558125183789787971, −8.201364572574261759026425774073, −7.12707352757895896260604746347, −6.65749829716567597812654887478, −5.57554421801130637442910554631, −4.54094572583977094282012943181, −3.93835278529017192382981231900, −3.22412714298363682565510952801, −1.39875847841893684697161315665,
0.31977115304105134625154788386, 1.53594362494527762656225120554, 2.77954387174995072395431218013, 4.03232827394358621798409958550, 5.02400551700898637918425369742, 5.86719616877824092674159644256, 6.35144895167287782233063438713, 7.69566029694041409066510871705, 8.075323914613659037820864216106, 8.755913280025412819840764640875