| L(s) = 1 | + (−0.459 + 0.134i)2-s + (0.260 + 1.81i)3-s + (−1.48 + 0.957i)4-s + (−0.345 + 0.755i)5-s + (−0.364 − 0.797i)6-s + (1.04 − 0.306i)7-s + (1.18 − 1.36i)8-s + (−0.337 + 0.0991i)9-s + (0.0566 − 0.393i)10-s + (1.19 − 2.61i)11-s + (−2.12 − 2.44i)12-s + (2.87 + 3.31i)13-s + (−0.438 + 0.281i)14-s + (−1.45 − 0.428i)15-s + (1.11 − 2.43i)16-s + (−2.76 − 1.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.324 + 0.0954i)2-s + (0.150 + 1.04i)3-s + (−0.744 + 0.478i)4-s + (−0.154 + 0.337i)5-s + (−0.148 − 0.325i)6-s + (0.394 − 0.115i)7-s + (0.418 − 0.482i)8-s + (−0.112 + 0.0330i)9-s + (0.0179 − 0.124i)10-s + (0.360 − 0.788i)11-s + (−0.612 − 0.707i)12-s + (0.797 + 0.920i)13-s + (−0.117 + 0.0752i)14-s + (−0.376 − 0.110i)15-s + (0.277 − 0.608i)16-s + (−0.670 − 0.430i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.574315 + 0.465338i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.574315 + 0.465338i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (-7.85 - 2.31i)T \) |
| good | 2 | \( 1 + (0.459 - 0.134i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (-0.260 - 1.81i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (0.345 - 0.755i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-1.04 + 0.306i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.19 + 2.61i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-2.87 - 3.31i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.76 + 1.77i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.40 + 1.00i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (0.912 + 6.34i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 + 1.97T + 29T^{2} \) |
| 31 | \( 1 + (1.76 - 2.03i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 - 8.07T + 37T^{2} \) |
| 41 | \( 1 + (2.28 + 1.46i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (6.51 + 4.18i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (1.56 + 10.8i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (-6.30 + 4.05i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (6.97 - 8.05i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (3.84 + 8.41i)T + (-39.9 + 46.1i)T^{2} \) |
| 71 | \( 1 + (1.45 - 0.937i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.608 + 1.33i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (10.1 + 11.7i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (3.35 - 7.35i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (2.19 - 15.2i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 - 1.03T + 97T^{2} \) |
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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
−15.08111461086647854798739847285, −14.11202373298083776198266508538, −13.05502721727355658201412736588, −11.43230528105687467403753512954, −10.46943537512090280925577759619, −9.137888340156706162874634868212, −8.518869370581543061819966647311, −6.79551034006819437049989593213, −4.67050108988407103018865810316, −3.67925132915766215922594314522,
1.55511832201145678556558369493, 4.46556961928454067525794577897, 6.11502036072175332347710443003, 7.75003749597622371150081856035, 8.604166944713344541906020915802, 9.931415833351849496385381879860, 11.24284097956735401345101377385, 12.74891132419557578256651703801, 13.23700272881421170947026607500, 14.47602593784088764066540419767
