| 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 | \( 1 + (0.693 - 0.353i)T + (57.0 - 78.4i)T^{2} \) |
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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
−14.27785636250832343458106521692, −13.30930441710573781204889287504, −12.53189008558240327164330785497, −11.48140207035895153651492022095, −10.18092237424042958557447676860, −9.316731514749666605674995476845, −7.913115904540878915964046199170, −6.35926445101523695259233659890, −3.21989557739439177488018901190, −1.73400832489187765669701337363,
4.58722014779660444611556568449, 5.85480219575054147876157218945, 7.23211889083396910615378824502, 8.860759429996516773503136427394, 9.697446511173140863860605805022, 10.15264522018887306550417286917, 12.70761404959776702544799077261, 14.22180203959700526542440166660, 14.94285117375552812496358128966, 16.00329107519688313638553819147
