| L(s) = 1 | + (0.618 + 1.61i)3-s + i·5-s + 2i·7-s + (−2.23 + 2.00i)9-s − 2.47·11-s − 1.23·13-s + (−1.61 + 0.618i)15-s + 0.763i·17-s + 5.23i·19-s + (−3.23 + 1.23i)21-s + 0.472·23-s − 25-s + (−4.61 − 2.38i)27-s − 8.47i·29-s − 4.76i·31-s + ⋯ |
| L(s) = 1 | + (0.356 + 0.934i)3-s + 0.447i·5-s + 0.755i·7-s + (−0.745 + 0.666i)9-s − 0.745·11-s − 0.342·13-s + (−0.417 + 0.159i)15-s + 0.185i·17-s + 1.20i·19-s + (−0.706 + 0.269i)21-s + 0.0984·23-s − 0.200·25-s + (−0.888 − 0.458i)27-s − 1.57i·29-s − 0.855i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9422907736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9422907736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 - 1.61i)T \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 0.763iT - 17T^{2} \) |
| 19 | \( 1 - 5.23iT - 19T^{2} \) |
| 23 | \( 1 - 0.472T + 23T^{2} \) |
| 29 | \( 1 + 8.47iT - 29T^{2} \) |
| 31 | \( 1 + 4.76iT - 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 - 1.52iT - 41T^{2} \) |
| 43 | \( 1 - 9.70iT - 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 + 8.18iT - 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 1.52iT - 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.731013252674235107224384292080, −8.936021981826733656648280736208, −8.069794335266073431637470839999, −7.60877675813312021160365898275, −6.20289007885092285986903233434, −5.64609920965876104981226597546, −4.73109099786135824089141930353, −3.81462981556797426286736285475, −2.86556133120541789345981876709, −2.10627793849199155826054872299,
0.30949667441041139373548064271, 1.50540833196602630401450582519, 2.66240892412814673663876096987, 3.56218470095438494251460000964, 4.81451981152769045946451468919, 5.47071565102309374667598200922, 6.71450112736747976007333838852, 7.19919967409669767220143278140, 7.85505679417149756991644231506, 8.833784407893177634438954918430
