L(s) = 1 | + 2-s + (1.29 − 1.14i)3-s + 4-s + (−1.07 − 1.96i)5-s + (1.29 − 1.14i)6-s + 8-s + (0.375 − 2.97i)9-s + (−1.07 − 1.96i)10-s − 0.115i·11-s + (1.29 − 1.14i)12-s − 5.60·13-s + (−3.64 − 1.31i)15-s + 16-s − 1.39i·17-s + (0.375 − 2.97i)18-s − 1.52i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.750 − 0.661i)3-s + 0.5·4-s + (−0.479 − 0.877i)5-s + (0.530 − 0.467i)6-s + 0.353·8-s + (0.125 − 0.992i)9-s + (−0.339 − 0.620i)10-s − 0.0348i·11-s + (0.375 − 0.330i)12-s − 1.55·13-s + (−0.940 − 0.340i)15-s + 0.250·16-s − 0.337i·17-s + (0.0884 − 0.701i)18-s − 0.349i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.687480192\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.687480192\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.29 + 1.14i)T \) |
| 5 | \( 1 + (1.07 + 1.96i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.115iT - 11T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 + 1.39iT - 17T^{2} \) |
| 19 | \( 1 + 1.52iT - 19T^{2} \) |
| 23 | \( 1 - 6.58T + 23T^{2} \) |
| 29 | \( 1 + 8.97iT - 29T^{2} \) |
| 31 | \( 1 + 7.15iT - 31T^{2} \) |
| 37 | \( 1 + 1.70iT - 37T^{2} \) |
| 41 | \( 1 - 3.82T + 41T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 - 2.37iT - 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 3.32T + 59T^{2} \) |
| 61 | \( 1 - 5.12iT - 61T^{2} \) |
| 67 | \( 1 - 0.689iT - 67T^{2} \) |
| 71 | \( 1 - 6.77iT - 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 6.24iT - 83T^{2} \) |
| 89 | \( 1 - 6.83T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.429742915889111469585153991033, −8.151359164954972932958153489296, −7.67036337379184787303770616964, −6.94710968690445636949634554137, −5.92016294642158694898353904465, −4.84744805678801168631383835154, −4.23320528702361865049436166309, −3.02452823243683593991065718690, −2.20744406830336557599030757762, −0.74148159690587191546951119046,
2.07007363358315994276117279605, 3.05348257927296896750072673990, 3.59604773957979896740321102819, 4.74200956088740707115560921554, 5.27321043254893179922511076229, 6.74679924521238270476730660504, 7.23600577787377721291149721051, 8.056002494374715074632317229476, 9.008045927305112152502366383662, 9.882878008161361637155920780859