| L(s) = 1 | + (−1.72 − 2.05i)2-s + (−0.734 + 2.01i)3-s + (−0.906 + 5.13i)4-s + (1.11 − 1.93i)5-s + (5.41 − 1.97i)6-s + (2.41 + 1.39i)7-s + (7.48 − 4.32i)8-s + (−1.23 − 1.03i)9-s + (−5.91 + 1.04i)10-s + (1.18 + 2.06i)11-s + (−9.70 − 5.60i)12-s + (−0.0138 − 0.0380i)13-s + (−1.29 − 7.36i)14-s + (3.08 + 3.67i)15-s + (−12.0 − 4.37i)16-s + (1.16 + 1.39i)17-s + ⋯ |
| L(s) = 1 | + (−1.22 − 1.45i)2-s + (−0.423 + 1.16i)3-s + (−0.453 + 2.56i)4-s + (0.500 − 0.865i)5-s + (2.21 − 0.805i)6-s + (0.910 + 0.525i)7-s + (2.64 − 1.52i)8-s + (−0.410 − 0.344i)9-s + (−1.87 + 0.329i)10-s + (0.358 + 0.621i)11-s + (−2.80 − 1.61i)12-s + (−0.00384 − 0.0105i)13-s + (−0.347 − 1.96i)14-s + (0.796 + 0.949i)15-s + (−3.00 − 1.09i)16-s + (0.283 + 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.561873 - 0.107339i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.561873 - 0.107339i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.11 + 1.93i)T \) |
| 19 | \( 1 + (-4.07 - 1.54i)T \) |
| good | 2 | \( 1 + (1.72 + 2.05i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (0.734 - 2.01i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-2.41 - 1.39i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.18 - 2.06i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0138 + 0.0380i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 1.39i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (2.50 + 0.441i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.25 + 1.89i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.44 + 2.49i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.227iT - 37T^{2} \) |
| 41 | \( 1 + (7.55 + 2.74i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (5.05 - 0.891i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.11 + 8.48i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-5.62 - 0.992i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (8.89 - 7.46i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.795 + 4.51i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.11 + 3.71i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.34 + 7.65i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.45 - 3.99i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-10.9 - 3.97i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (3.87 + 2.23i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-14.4 + 5.26i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (3.59 + 4.28i)T + (-16.8 + 95.5i)T^{2} \) |
| 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
−13.56003197751000977621949372196, −12.18222663006398722435226459301, −11.68908729181909471525109223142, −10.46174211699317792569989775120, −9.766655215473468533453920047710, −8.964729602319217960397733746736, −7.912428355443829095583522651199, −5.19208912522621386079103372582, −3.98732034427789184069776946322, −1.81391661785676163197941626116,
1.34892844592127533335860448870, 5.42562725063181456306692085924, 6.49063122771182256620718462097, 7.25992181143955238367807224655, 8.042528175975719475113658989529, 9.459744096303289068861030624545, 10.63274621491732385084807118404, 11.61263871788376772445748012010, 13.66556646222123170315348069013, 14.10920757948115062886165363216
