L(s) = 1 | + (1.06 + 1.06i)2-s + (0.0150 + 1.73i)3-s + 0.285i·4-s + (2.03 − 0.933i)5-s + (−1.83 + 1.86i)6-s + (1.83 − 1.83i)8-s + (−2.99 + 0.0520i)9-s + (3.16 + 1.17i)10-s − 0.914i·11-s + (−0.493 + 0.00428i)12-s + (3.07 + 3.07i)13-s + (1.64 + 3.50i)15-s + 4.48·16-s + (0.850 + 0.850i)17-s + (−3.26 − 3.15i)18-s + 6.87i·19-s + ⋯ |
L(s) = 1 | + (0.755 + 0.755i)2-s + (0.00867 + 0.999i)3-s + 0.142i·4-s + (0.908 − 0.417i)5-s + (−0.749 + 0.762i)6-s + (0.648 − 0.648i)8-s + (−0.999 + 0.0173i)9-s + (1.00 + 0.371i)10-s − 0.275i·11-s + (−0.142 + 0.00123i)12-s + (0.854 + 0.854i)13-s + (0.425 + 0.905i)15-s + 1.12·16-s + (0.206 + 0.206i)17-s + (−0.768 − 0.742i)18-s + 1.57i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02692 + 1.80735i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02692 + 1.80735i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0150 - 1.73i)T \) |
| 5 | \( 1 + (-2.03 + 0.933i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.06 - 1.06i)T + 2iT^{2} \) |
| 11 | \( 1 + 0.914iT - 11T^{2} \) |
| 13 | \( 1 + (-3.07 - 3.07i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.850 - 0.850i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.87iT - 19T^{2} \) |
| 23 | \( 1 + (1.38 - 1.38i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 - 4.63T + 31T^{2} \) |
| 37 | \( 1 + (-0.567 + 0.567i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.922iT - 41T^{2} \) |
| 43 | \( 1 + (4.80 + 4.80i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.41 + 7.41i)T + 47iT^{2} \) |
| 53 | \( 1 + (-7.79 + 7.79i)T - 53iT^{2} \) |
| 59 | \( 1 + 9.88T + 59T^{2} \) |
| 61 | \( 1 + 1.06T + 61T^{2} \) |
| 67 | \( 1 + (5.00 - 5.00i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.557iT - 71T^{2} \) |
| 73 | \( 1 + (1.54 + 1.54i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.03iT - 79T^{2} \) |
| 83 | \( 1 + (-2.38 + 2.38i)T - 83iT^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + (1.58 - 1.58i)T - 97iT^{2} \) |
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
−10.24082135285503876350957183609, −9.923228141479592051857823661540, −8.883242952508167527661992782575, −8.093570576042217627613773906222, −6.62361565297320099283153361413, −5.90107334514144715051739436479, −5.34871018846601933356877121101, −4.32388154899016949095498528981, −3.53304991850672220243594222055, −1.66171997879247358416385934267,
1.36855993159039977686223093894, 2.57170408231199942456328789625, 3.16590347944261237540450944254, 4.71608270105165512260126853929, 5.69630302567408490498312277814, 6.53443358623149156783299395985, 7.51078305194462227579911908206, 8.376291810598105168038384554359, 9.400591880843830371346601002466, 10.58725421191516765970379322045