L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + i·5-s + (−1.5 + 0.866i)7-s − 0.999i·8-s + (0.5 − 0.866i)10-s + (−2.59 − 1.5i)11-s + (3.5 − 0.866i)13-s + 1.73·14-s + (−0.5 + 0.866i)16-s + (4.5 − 2.59i)19-s + (−0.866 + 0.499i)20-s + (1.5 + 2.59i)22-s + (−1.73 + 3i)23-s − 25-s + (−3.46 − i)26-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (−0.566 + 0.327i)7-s − 0.353i·8-s + (0.158 − 0.273i)10-s + (−0.783 − 0.452i)11-s + (0.970 − 0.240i)13-s + 0.462·14-s + (−0.125 + 0.216i)16-s + (1.03 − 0.596i)19-s + (−0.193 + 0.111i)20-s + (0.319 + 0.553i)22-s + (−0.361 + 0.625i)23-s − 0.200·25-s + (−0.679 − 0.196i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9984604276\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9984604276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-3.5 + 0.866i)T \) |
good | 7 | \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.46 + 6i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + (-7.5 - 4.33i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.19 - 3i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + (10.3 - 6i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.19 + 3i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (-12.9 - 7.5i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6 - 3.46i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.670716998525062474296023510104, −9.355661193052359836038954776649, −8.121656299923090509872913887686, −7.74455657077361876497136729216, −6.48040843868636936986873011333, −5.93591200514890429025625800234, −4.65088387459145165618235996187, −3.21100186546361693844877729837, −2.81706533892856470132188510100, −1.13799281355123868310687181497,
0.62533679537498635433639099711, 2.07041439750048115798518459712, 3.46179666406888936734040917164, 4.55815013585362866585935570210, 5.65860325515406083485264027347, 6.33251606732403172073131024762, 7.42093536262592766782674520389, 7.950459722789299333639445937846, 8.932452194567900184339047242677, 9.571298852525287762854224248979