L(s) = 1 | + (−0.965 + 0.258i)2-s + (−0.965 − 0.258i)3-s + (0.866 − 0.499i)4-s + (−0.965 − 0.258i)5-s + 6-s + (1.22 − 0.707i)7-s + (−0.707 + 0.707i)8-s + 10-s + (1 − i)11-s + (−0.965 + 0.258i)12-s + (−0.258 + 0.965i)13-s + (−0.999 + i)14-s + (0.866 + 0.499i)15-s + (0.500 − 0.866i)16-s + (1.22 + 0.707i)17-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (−0.965 − 0.258i)3-s + (0.866 − 0.499i)4-s + (−0.965 − 0.258i)5-s + 6-s + (1.22 − 0.707i)7-s + (−0.707 + 0.707i)8-s + 10-s + (1 − i)11-s + (−0.965 + 0.258i)12-s + (−0.258 + 0.965i)13-s + (−0.999 + i)14-s + (0.866 + 0.499i)15-s + (0.500 − 0.866i)16-s + (1.22 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5677846884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5677846884\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 + (0.965 + 0.258i)T \) |
| 37 | \( 1 + (-0.965 - 0.258i)T \) |
good | 3 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1 + i)T - iT^{2} \) |
| 13 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.487168608126153118419266617094, −8.293059237528876855958151472714, −7.53112870771201978011175033390, −6.60314592631037738986353478204, −6.17720291713355646290266156959, −5.17351210696649550684570491952, −4.30934794166149182486208405394, −3.35795591282113823931740057784, −1.55555278687904423799558057748, −0.922077117190105643590034155688,
0.833364628418546272467392375680, 2.18044004169467190790894783262, 3.19170597788493676362576017763, 4.35423999224204191472718735476, 5.13274903674177810373480056345, 5.90059838009391899719373129729, 6.96382847565140765453488165159, 7.64324863281709399031646137171, 8.049257773395797543439718236887, 9.070211407862398044239193803171