L(s) = 1 | + (0.866 + 0.5i)3-s + (−1.73 − 2i)7-s + (−1 − 1.73i)9-s + (1.5 − 2.59i)11-s − 2i·13-s + (−2.59 − 1.5i)17-s + (−0.5 − 0.866i)19-s + (−0.499 − 2.59i)21-s + (−2.59 + 1.5i)23-s − 5i·27-s + 6·29-s + (3.5 − 6.06i)31-s + (2.59 − 1.5i)33-s + (−0.866 + 0.5i)37-s + (1 − 1.73i)39-s + ⋯ |
L(s) = 1 | + (0.499 + 0.288i)3-s + (−0.654 − 0.755i)7-s + (−0.333 − 0.577i)9-s + (0.452 − 0.783i)11-s − 0.554i·13-s + (−0.630 − 0.363i)17-s + (−0.114 − 0.198i)19-s + (−0.109 − 0.566i)21-s + (−0.541 + 0.312i)23-s − 0.962i·27-s + 1.11·29-s + (0.628 − 1.08i)31-s + (0.452 − 0.261i)33-s + (−0.142 + 0.0821i)37-s + (0.160 − 0.277i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06178 - 0.879988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06178 - 0.879988i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
good | 3 | \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (7.79 - 4.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.59 - 1.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.06 - 3.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (0.866 + 0.5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−10.07129323016138876527103486133, −9.463916053288079298959744041567, −8.601890673690162313869960100227, −7.78383071455577516978756509009, −6.59397572795414659931038445589, −6.00337271504048701634095107857, −4.50534004762689415947944390747, −3.58943943013354949596307518958, −2.74432547150436190087870837135, −0.67332281596585373974923149882,
1.89529636746624568587643586417, 2.78977830424068671079950438877, 4.11086253759040976385742962966, 5.21655766996772904948754582958, 6.38719467063778965646577524395, 7.03205833141030878966178964370, 8.277978880480883859635194670734, 8.775295305273272733110146387051, 9.702650609091851689372898489662, 10.51158942955886414524007769717