| L(s) = 1 | + (−1.67 + 2.09i)2-s + (−0.149 − 0.989i)3-s + (−1.15 − 5.08i)4-s + (−2.44 − 3.58i)5-s + (2.32 + 1.34i)6-s + (0.298 − 0.172i)7-s + (7.77 + 3.74i)8-s + (1.90 − 0.588i)9-s + (11.6 + 0.871i)10-s + (3.18 + 0.727i)11-s + (−4.85 + 1.90i)12-s + (−0.285 − 3.80i)13-s + (−0.137 + 0.914i)14-s + (−3.18 + 2.95i)15-s + (−11.4 + 5.53i)16-s + (4.05 + 0.726i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 + 1.48i)2-s + (−0.0861 − 0.571i)3-s + (−0.579 − 2.54i)4-s + (−1.09 − 1.60i)5-s + (0.950 + 0.548i)6-s + (0.112 − 0.0650i)7-s + (2.74 + 1.32i)8-s + (0.636 − 0.196i)9-s + (3.67 + 0.275i)10-s + (0.961 + 0.219i)11-s + (−1.40 + 0.550i)12-s + (−0.0791 − 1.05i)13-s + (−0.0368 + 0.244i)14-s + (−0.822 + 0.763i)15-s + (−2.87 + 1.38i)16-s + (0.984 + 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.451338 - 0.400694i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451338 - 0.400694i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-4.05 - 0.726i)T \) |
| 43 | \( 1 + (3.70 + 5.40i)T \) |
| good | 2 | \( 1 + (1.67 - 2.09i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (0.149 + 0.989i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (2.44 + 3.58i)T + (-1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (-0.298 + 0.172i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.18 - 0.727i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.285 + 3.80i)T + (-12.8 + 1.93i)T^{2} \) |
| 19 | \( 1 + (3.96 + 1.22i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (-4.78 + 5.15i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.840 + 5.57i)T + (-27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (-3.71 + 1.45i)T + (22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (-1.26 - 0.731i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.16 - 1.72i)T + (9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-1.12 - 4.91i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.473 + 6.31i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (6.39 - 3.08i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (11.7 + 4.60i)T + (44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-9.67 - 2.98i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (-0.992 - 1.06i)T + (-5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (3.26 - 0.244i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (2.78 - 1.60i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.16 + 1.08i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (-5.91 + 0.891i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (11.8 + 2.71i)T + (87.3 + 42.0i)T^{2} \) |
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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
−9.705007642352276841755391371760, −9.049914340033085370911410350251, −8.094638280414314933431653681622, −7.929927395494118729565091512558, −6.92604423030354538338414409758, −6.06962919053200149424509344395, −4.94811161186226670788866126208, −4.22635014454869591533623460486, −1.28757951458888957154296530759, −0.57812030604868092557947250722,
1.57984654185268295883383577332, 3.02948465076652423903360299218, 3.68868506343822089271834614541, 4.43811483008888242230914746482, 6.72670520623297548134858213431, 7.34177255926041274943116761750, 8.214558241475108924961576838611, 9.235539359322520900144679440215, 9.901950972415855885577073555018, 10.75262737473789965334753219422
