| L(s) = 1 | + (0.247 − 2.35i)2-s + (−3.06 − 0.651i)3-s + (−3.51 − 0.748i)4-s + (−0.346 + 2.20i)5-s + (−2.29 + 7.05i)6-s + (−0.0714 − 2.64i)7-s + (−1.16 + 3.59i)8-s + (6.23 + 2.77i)9-s + (5.11 + 1.36i)10-s + (−2.55 + 1.13i)11-s + (10.3 + 4.58i)12-s + (−3.53 + 2.56i)13-s + (−6.24 − 0.486i)14-s + (2.50 − 6.54i)15-s + (1.59 + 0.711i)16-s + (−0.495 − 0.550i)17-s + ⋯ |
| L(s) = 1 | + (0.174 − 1.66i)2-s + (−1.77 − 0.376i)3-s + (−1.75 − 0.374i)4-s + (−0.155 + 0.987i)5-s + (−0.935 + 2.88i)6-s + (−0.0269 − 0.999i)7-s + (−0.413 + 1.27i)8-s + (2.07 + 0.925i)9-s + (1.61 + 0.430i)10-s + (−0.770 + 0.342i)11-s + (2.97 + 1.32i)12-s + (−0.979 + 0.711i)13-s + (−1.66 − 0.129i)14-s + (0.646 − 1.69i)15-s + (0.399 + 0.177i)16-s + (−0.120 − 0.133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0987341 + 0.0841484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0987341 + 0.0841484i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.346 - 2.20i)T \) |
| 7 | \( 1 + (0.0714 + 2.64i)T \) |
| good | 2 | \( 1 + (-0.247 + 2.35i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (3.06 + 0.651i)T + (2.74 + 1.22i)T^{2} \) |
| 11 | \( 1 + (2.55 - 1.13i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (3.53 - 2.56i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.495 + 0.550i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (2.77 - 0.589i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.251 + 2.39i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (0.498 + 1.53i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.24 + 1.37i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (6.57 + 2.92i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (-2.82 + 2.05i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.22T + 43T^{2} \) |
| 47 | \( 1 + (-0.246 + 0.274i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-9.38 - 1.99i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (0.658 + 6.26i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.38 + 13.1i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (5.66 + 6.29i)T + (-7.00 + 66.6i)T^{2} \) |
| 71 | \( 1 + (-3.83 - 11.8i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.95 - 0.871i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-2.50 + 2.77i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.181 + 0.557i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.30 + 12.3i)T + (-87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (-0.798 - 2.45i)T + (-78.4 + 57.0i)T^{2} \) |
| 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
−11.80346209056986475301217734284, −10.94753787510192690232693635601, −10.51297609410772855576668319962, −9.801163377188229968654314897093, −7.43916249775580548874455565634, −6.61930946970362450090418316136, −5.02792065450824852578189768342, −4.03583754494808436646534535669, −2.13588263420970733722685490521, −0.13448996156477137643015826083,
4.51246065757652649745352219110, 5.41591018621640926081710750454, 5.66927928782859661596609252182, 6.97546612894511382562192611819, 8.169491841058151058183845347505, 9.234940583988785060515442661566, 10.45573190916095595219649868012, 11.77708875435548574263100042276, 12.55997047391440228557344608826, 13.30252875191837859816289385572
