| L(s) = 1 | + (−1.15 + 2.25i)2-s + (0.313 + 1.98i)3-s + (−2.60 − 3.58i)4-s + (1.91 − 1.15i)5-s + (−4.83 − 1.57i)6-s + (−1.78 − 0.282i)7-s + (6.07 − 0.962i)8-s + (−0.969 + 0.315i)9-s + (0.412 + 5.65i)10-s + (1.72 + 2.83i)11-s + (6.27 − 6.27i)12-s + (−2.00 − 1.02i)13-s + (2.69 − 3.70i)14-s + (2.89 + 3.42i)15-s + (−2.08 + 6.41i)16-s + (2.99 − 1.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.813 + 1.59i)2-s + (0.181 + 1.14i)3-s + (−1.30 − 1.79i)4-s + (0.855 − 0.517i)5-s + (−1.97 − 0.641i)6-s + (−0.674 − 0.106i)7-s + (2.14 − 0.340i)8-s + (−0.323 + 0.105i)9-s + (0.130 + 1.78i)10-s + (0.519 + 0.854i)11-s + (1.81 − 1.81i)12-s + (−0.557 − 0.284i)13-s + (0.720 − 0.991i)14-s + (0.746 + 0.884i)15-s + (−0.521 + 1.60i)16-s + (0.726 − 0.370i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.216512 + 0.595037i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.216512 + 0.595037i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.91 + 1.15i)T \) |
| 11 | \( 1 + (-1.72 - 2.83i)T \) |
| good | 2 | \( 1 + (1.15 - 2.25i)T + (-1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (-0.313 - 1.98i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (1.78 + 0.282i)T + (6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (2.00 + 1.02i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.99 + 1.52i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (1.05 + 0.767i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.16 + 3.16i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.37 - 1.72i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.23 + 3.81i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.501 + 3.16i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.766 + 1.05i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.55 + 4.55i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.62 - 1.20i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (2.89 - 5.67i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-7.28 - 10.0i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.85 + 2.87i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (2.46 - 2.46i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.09 - 6.43i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.47 - 9.32i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.353 - 1.08i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.75 + 9.32i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + 3.85iT - 89T^{2} \) |
| 97 | \( 1 + (0.693 + 0.353i)T + (57.0 + 78.4i)T^{2} \) |
| 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
−16.00329107519688313638553819147, −14.94285117375552812496358128966, −14.22180203959700526542440166660, −12.70761404959776702544799077261, −10.15264522018887306550417286917, −9.697446511173140863860605805022, −8.860759429996516773503136427394, −7.23211889083396910615378824502, −5.85480219575054147876157218945, −4.58722014779660444611556568449,
1.73400832489187765669701337363, 3.21989557739439177488018901190, 6.35926445101523695259233659890, 7.913115904540878915964046199170, 9.316731514749666605674995476845, 10.18092237424042958557447676860, 11.48140207035895153651492022095, 12.53189008558240327164330785497, 13.30930441710573781204889287504, 14.27785636250832343458106521692
