| L(s) = 1 | + (−1.26 + 0.634i)2-s + (0.968 − 0.647i)3-s + (1.19 − 1.60i)4-s + (0.0404 − 0.203i)5-s + (−0.814 + 1.43i)6-s + (2.58 − 0.514i)7-s + (−0.494 + 2.78i)8-s + (−0.628 + 1.51i)9-s + (0.0777 + 0.282i)10-s + (0.166 − 0.249i)11-s + (0.120 − 2.32i)12-s + (−2.27 − 2.27i)13-s + (−2.94 + 2.29i)14-s + (−0.0923 − 0.223i)15-s + (−1.14 − 3.83i)16-s + (−3.65 + 1.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.893 + 0.448i)2-s + (0.559 − 0.373i)3-s + (0.597 − 0.801i)4-s + (0.0180 − 0.0908i)5-s + (−0.332 + 0.584i)6-s + (0.978 − 0.194i)7-s + (−0.174 + 0.984i)8-s + (−0.209 + 0.505i)9-s + (0.0245 + 0.0893i)10-s + (0.0503 − 0.0753i)11-s + (0.0348 − 0.671i)12-s + (−0.632 − 0.632i)13-s + (−0.787 + 0.612i)14-s + (−0.0238 − 0.0575i)15-s + (−0.285 − 0.958i)16-s + (−0.887 + 0.461i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.754047 + 0.0251797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.754047 + 0.0251797i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.634i)T \) |
| 17 | \( 1 + (3.65 - 1.90i)T \) |
| good | 3 | \( 1 + (-0.968 + 0.647i)T + (1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (-0.0404 + 0.203i)T + (-4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-2.58 + 0.514i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.166 + 0.249i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (2.27 + 2.27i)T + 13iT^{2} \) |
| 19 | \( 1 + (4.06 - 1.68i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (5.73 + 3.83i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-3.83 - 0.762i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-3.47 - 5.19i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (4.85 + 7.26i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-1.40 - 7.05i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (4.20 + 1.74i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-3.51 + 3.51i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.21 + 5.35i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.44 + 10.7i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-5.17 + 1.02i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + (-7.26 + 4.85i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (0.991 - 4.98i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (6.48 - 9.71i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (2.51 + 6.06i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.73 + 6.73i)T - 89iT^{2} \) |
| 97 | \( 1 + (1.95 + 0.389i)T + (89.6 + 37.1i)T^{2} \) |
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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
−14.68800913582560185764741642506, −14.16196053133989496043387559412, −12.63960806540135416963693409956, −11.11088312919360162881696390885, −10.25301541201460318759249369583, −8.532884988624309459966527904041, −8.121656853095432933472103574884, −6.74880295417673543533409339547, −5.02387373777189678214406350568, −2.12209818536780574077729955009,
2.40725790165054802334953924164, 4.31186522712876485061229620705, 6.69578098668890872478346889084, 8.182378166218362128828769503523, 9.016137873825263885530202048114, 10.09142564840817704228299217903, 11.38415460443841046539393754449, 12.16270031505705329380058598211, 13.80532979584509968777145956327, 14.94174637306727265776886888146
