L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s + 0.999i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s − i·13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.499i)18-s + (−0.5 + 0.866i)19-s + 0.999i·22-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.499i)28-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s + 0.999i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s − i·13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.866 − 0.499i)18-s + (−0.5 + 0.866i)19-s + 0.999i·22-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (0.866 − 0.499i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.856041779\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.856041779\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−9.967066307115873797552771171660, −8.914721527312693272260308244954, −7.937339122557652332037629190499, −7.16131993941169077278009588275, −6.69366977624220667306758680083, −5.69772431540682343415447595872, −4.69796863583890074529305885822, −3.89944864340438923751048250794, −3.22092628046111606104817532526, −1.58105235139067307344594910014,
1.64375435082450942846861992013, 2.56916313280053855079446431611, 3.60539332497162277590541150234, 4.75170224060472153993523193261, 5.21939050070547900073886043687, 6.44410319688415009718417845928, 6.80149828697297067313718488760, 8.203925436163861706452882374116, 9.006632373163544411120614787731, 9.726579668190755514421777113304