L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.169 − 1.72i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−1.33 + 1.09i)6-s + 1.82·7-s + (0.707 − 0.707i)8-s + (−2.94 − 0.584i)9-s − 1.00·10-s + 5.15·11-s + (1.72 + 0.169i)12-s + (−3.16 − 3.16i)13-s + (−1.29 − 1.29i)14-s + (−1.09 − 1.33i)15-s − 1.00·16-s + (−2.98 + 2.98i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.0979 − 0.995i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (−0.546 + 0.448i)6-s + 0.690·7-s + (0.250 − 0.250i)8-s + (−0.980 − 0.194i)9-s − 0.316·10-s + 1.55·11-s + (0.497 + 0.0489i)12-s + (−0.878 − 0.878i)13-s + (−0.345 − 0.345i)14-s + (−0.283 − 0.345i)15-s − 0.250·16-s + (−0.724 + 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.342454427\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.342454427\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.169 + 1.72i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (-0.159 + 6.08i)T \) |
good | 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 - 5.15T + 11T^{2} \) |
| 13 | \( 1 + (3.16 + 3.16i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.98 - 2.98i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.92 + 2.92i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.68 + 3.68i)T - 23iT^{2} \) |
| 29 | \( 1 + (-4.89 - 4.89i)T + 29iT^{2} \) |
| 31 | \( 1 + (-6.21 + 6.21i)T - 31iT^{2} \) |
| 41 | \( 1 + 0.816T + 41T^{2} \) |
| 43 | \( 1 + (2.69 + 2.69i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.46iT - 47T^{2} \) |
| 53 | \( 1 + 12.1iT - 53T^{2} \) |
| 59 | \( 1 + (0.207 - 0.207i)T - 59iT^{2} \) |
| 61 | \( 1 + (8.45 - 8.45i)T - 61iT^{2} \) |
| 67 | \( 1 + 12.0iT - 67T^{2} \) |
| 71 | \( 1 - 11.3iT - 71T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 + (-5.32 - 5.32i)T + 79iT^{2} \) |
| 83 | \( 1 - 14.3iT - 83T^{2} \) |
| 89 | \( 1 + (6.37 + 6.37i)T + 89iT^{2} \) |
| 97 | \( 1 + (8.32 + 8.32i)T + 97iT^{2} \) |
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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
−9.326520290218264845045284767728, −8.597112046830022291827844152833, −8.154455206162572224905694306245, −6.95941204788821851968988136846, −6.48018535719134209363958358897, −5.19236008460313352138605198311, −4.14660025940369690917785968899, −2.72651149595209899080992547849, −1.82334678190894134847119858953, −0.72169457131654457950426644993,
1.60925667783646943483333358090, 2.97319001409912494650104030184, 4.45118943463429961382354801508, 4.77659635896162942145984859092, 6.16884726538893741728683752520, 6.71830722980200085394380326705, 7.82766862266770189868602897430, 8.775910654983400569548111763188, 9.321853920266664337581380024416, 9.934035410036007404625535332060