L(s) = 1 | + (1.02 − 1.77i)2-s + (0.5 − 0.866i)3-s + (−1.09 − 1.90i)4-s + 3.35·5-s + (−1.02 − 1.77i)6-s + (−1.12 − 1.94i)7-s − 0.405·8-s + (−0.499 − 0.866i)9-s + (3.43 − 5.95i)10-s + (−2.46 + 4.27i)11-s − 2.19·12-s − 4.60·14-s + (1.67 − 2.90i)15-s + (1.78 − 3.08i)16-s + (−0.455 − 0.789i)17-s − 2.04·18-s + ⋯ |
L(s) = 1 | + (0.724 − 1.25i)2-s + (0.288 − 0.499i)3-s + (−0.549 − 0.951i)4-s + 1.50·5-s + (−0.418 − 0.724i)6-s + (−0.424 − 0.735i)7-s − 0.143·8-s + (−0.166 − 0.288i)9-s + (1.08 − 1.88i)10-s + (−0.744 + 1.28i)11-s − 0.634·12-s − 1.23·14-s + (0.433 − 0.750i)15-s + (0.445 − 0.771i)16-s + (−0.110 − 0.191i)17-s − 0.482·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25119 - 2.36616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25119 - 2.36616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.02 + 1.77i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 + (1.12 + 1.94i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.46 - 4.27i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.455 + 0.789i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.90 - 3.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.01 - 1.75i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.96 + 3.41i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + (4.40 - 7.62i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.46 - 6.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.14 - 1.97i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 - 0.542T + 53T^{2} \) |
| 59 | \( 1 + (-2.35 - 4.08i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.83 + 3.18i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.760 + 1.31i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.18 + 2.05i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 2.30T + 83T^{2} \) |
| 89 | \( 1 + (-5.02 + 8.71i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.06 - 13.9i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.39842787228495702583072143242, −10.05731846616667382092406352121, −9.355875033640024465656369038901, −7.77189529993606639231076936315, −6.88060921120070803746993161214, −5.68667384737958148988396868539, −4.73937254203500709960744186525, −3.44348680073405723937489088509, −2.32161877120366555697115602025, −1.53504347334447203548324367310,
2.31739845005765211038363744270, 3.55375818503327849024939969543, 5.17607176035445182224520503183, 5.56707934906715591583436472021, 6.28948616808406321870993245873, 7.36388976512017438375324763397, 8.686171390478885916671202472515, 9.090956536021947779279112073407, 10.29676532875903782913572508391, 10.94566125643218931045478215651