L(s) = 1 | + (0.866 + 0.5i)2-s + (−2.59 + 1.5i)3-s + (0.499 + 0.866i)4-s + (1.23 + 1.86i)5-s − 3·6-s + (0.866 − 2.5i)7-s + 0.999i·8-s + (3 − 5.19i)9-s + (0.133 + 2.23i)10-s + (−2.59 − 1.50i)12-s − 2i·13-s + (2 − 1.73i)14-s + (−6 − 3i)15-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + (5.19 − 3i)18-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−1.49 + 0.866i)3-s + (0.249 + 0.433i)4-s + (0.550 + 0.834i)5-s − 1.22·6-s + (0.327 − 0.944i)7-s + 0.353i·8-s + (1 − 1.73i)9-s + (0.0423 + 0.705i)10-s + (−0.749 − 0.433i)12-s − 0.554i·13-s + (0.534 − 0.462i)14-s + (−1.54 − 0.774i)15-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + (1.22 − 0.707i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.708696 + 0.573475i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.708696 + 0.573475i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-1.23 - 1.86i)T \) |
| 7 | \( 1 + (-0.866 + 2.5i)T \) |
good | 3 | \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 + (-6.92 - 4i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−15.01378994561969518513406450315, −14.04688400236946830188324989970, −12.74016830980429047250725311396, −11.39444897057865450990211041701, −10.70567023931131578997706477842, −9.806660943830376443911478166377, −7.42171420125352958769405301091, −6.21924130930249366732776460977, −5.22625661068966867787152024155, −3.81770243724999369473839296429,
1.72010951584794250908056878961, 4.94425525517896381904069506670, 5.67341944065894258133949738608, 6.84056006356308735358610205757, 8.745418404231952740762255374242, 10.39247182324690179314206702678, 11.61538378737480570837057754073, 12.28117800541804475435058764838, 12.97293312071078694527113425833, 14.12965696730763055122141290614