L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1.89 + 1.18i)5-s + (0.705 + 0.705i)7-s + (−0.707 − 0.707i)8-s + (−0.498 + 2.17i)10-s − 1.32·11-s + (0.741 + 0.741i)13-s + 0.997·14-s − 1.00·16-s + (−2.17 − 2.17i)17-s + (−0.939 − 4.25i)19-s + (1.18 + 1.89i)20-s + (−0.939 + 0.939i)22-s + (1.08 − 1.08i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.846 + 0.531i)5-s + (0.266 + 0.266i)7-s + (−0.250 − 0.250i)8-s + (−0.157 + 0.689i)10-s − 0.400·11-s + (0.205 + 0.205i)13-s + 0.266·14-s − 0.250·16-s + (−0.527 − 0.527i)17-s + (−0.215 − 0.976i)19-s + (0.265 + 0.423i)20-s + (−0.200 + 0.200i)22-s + (0.225 − 0.225i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9686226102\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9686226102\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (1.89 - 1.18i)T \) |
| 19 | \( 1 + (0.939 + 4.25i)T \) |
good | 7 | \( 1 + (-0.705 - 0.705i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + (-0.741 - 0.741i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.17 + 2.17i)T + 17iT^{2} \) |
| 23 | \( 1 + (-1.08 + 1.08i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.95T + 29T^{2} \) |
| 31 | \( 1 + 5.95iT - 31T^{2} \) |
| 37 | \( 1 + (-1.33 + 1.33i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.531iT - 41T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.53i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.13 + 4.13i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.48 + 5.48i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.53T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 + (-1.99 + 1.99i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.71iT - 71T^{2} \) |
| 73 | \( 1 + (5.83 - 5.83i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + (-9.50 + 9.50i)T - 83iT^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + (1.11 - 1.11i)T - 97iT^{2} \) |
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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
−9.033023360240732978414296731422, −8.245321637348846442153056361860, −7.30303680898462332464741793107, −6.65957023922055000304441652830, −5.59829305281447942792835934921, −4.69940400939892193593455360641, −3.94631231597415116161813932789, −2.94843178448240105154028024579, −2.09897417510511110301470551778, −0.30486318574693969738578383909,
1.50298921547898647130248749420, 3.09476980305336010104786164255, 3.97599912943784893494779111473, 4.67901581771604037510350271120, 5.53787822240790237974565637920, 6.40973205069235435856485165017, 7.43885589731463958781797591734, 7.941316954934037938708294798516, 8.617368106341856896350482337554, 9.455227835361829636376515194073