| L(s) = 1 | + (0.576 − 1.64i)2-s + (−0.221 − 1.71i)3-s + (−0.816 − 0.651i)4-s + (−1.59 + 0.769i)5-s + (−2.95 − 0.624i)6-s + (1.54 + 1.94i)7-s + (1.41 − 0.887i)8-s + (−2.90 + 0.761i)9-s + (0.346 + 3.07i)10-s + (1.02 + 0.644i)11-s + (−0.937 + 1.54i)12-s + (−1.08 + 0.246i)13-s + (4.08 − 1.42i)14-s + (1.67 + 2.57i)15-s + (−1.11 − 4.87i)16-s + (2.39 − 2.39i)17-s + ⋯ |
| L(s) = 1 | + (0.407 − 1.16i)2-s + (−0.128 − 0.991i)3-s + (−0.408 − 0.325i)4-s + (−0.714 + 0.344i)5-s + (−1.20 − 0.255i)6-s + (0.584 + 0.733i)7-s + (0.499 − 0.313i)8-s + (−0.967 + 0.253i)9-s + (0.109 + 0.972i)10-s + (0.309 + 0.194i)11-s + (−0.270 + 0.446i)12-s + (−0.299 + 0.0683i)13-s + (1.09 − 0.382i)14-s + (0.432 + 0.664i)15-s + (−0.278 − 1.21i)16-s + (0.579 − 0.579i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.655705 - 0.903057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.655705 - 0.903057i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.221 + 1.71i)T \) |
| 29 | \( 1 + (-5.27 + 1.06i)T \) |
| good | 2 | \( 1 + (-0.576 + 1.64i)T + (-1.56 - 1.24i)T^{2} \) |
| 5 | \( 1 + (1.59 - 0.769i)T + (3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (-1.54 - 1.94i)T + (-1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (-1.02 - 0.644i)T + (4.77 + 9.91i)T^{2} \) |
| 13 | \( 1 + (1.08 - 0.246i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-2.39 + 2.39i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.482 + 0.0543i)T + (18.5 - 4.22i)T^{2} \) |
| 23 | \( 1 + (3.99 - 8.30i)T + (-14.3 - 17.9i)T^{2} \) |
| 31 | \( 1 + (9.19 + 3.21i)T + (24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (-4.30 - 6.84i)T + (-16.0 + 33.3i)T^{2} \) |
| 41 | \( 1 + (6.28 + 6.28i)T + 41iT^{2} \) |
| 43 | \( 1 + (1.18 + 3.39i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (2.28 - 3.63i)T + (-20.3 - 42.3i)T^{2} \) |
| 53 | \( 1 + (5.08 + 10.5i)T + (-33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + 2.84iT - 59T^{2} \) |
| 61 | \( 1 + (-0.299 + 2.65i)T + (-59.4 - 13.5i)T^{2} \) |
| 67 | \( 1 + (-11.9 - 2.72i)T + (60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (-1.90 - 8.34i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.41 + 0.494i)T + (57.0 - 45.5i)T^{2} \) |
| 79 | \( 1 + (-1.96 + 1.23i)T + (34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (6.97 + 5.56i)T + (18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (3.09 - 8.84i)T + (-69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (0.582 + 5.16i)T + (-94.5 + 21.5i)T^{2} \) |
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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
−13.59741625571193657711987592221, −12.48152979577404998883181591520, −11.60591273840178101874043445981, −11.42486045204338227450675379632, −9.721284699072335622221316319872, −8.045326318292592046392772309507, −7.11832491091570642162455142841, −5.31054290250738454993973648996, −3.45606877783753904470385023005, −1.91858484221092708211766830454,
4.02024501125438837456155527259, 4.87855095229862136536588042110, 6.24355845449368181664892191245, 7.72814426763827721213721996951, 8.570081388360955177891813422193, 10.26792631794624583690889691287, 11.15267157708493306563181707157, 12.44013814499794363326932729872, 14.17938968630809028793504162248, 14.52868345928280618361821323106
