| L(s) = 1 | + (1.25 − 0.368i)2-s + (−0.385 − 2.68i)3-s + (−0.239 + 0.153i)4-s + (−1.35 + 2.95i)5-s + (−1.47 − 3.22i)6-s + (4.00 − 1.17i)7-s + (−1.95 + 2.26i)8-s + (−4.17 + 1.22i)9-s + (−0.605 + 4.21i)10-s + (−0.0622 + 0.136i)11-s + (0.505 + 0.583i)12-s + (−0.628 − 0.725i)13-s + (4.60 − 2.95i)14-s + (8.45 + 2.48i)15-s + (−1.39 + 3.04i)16-s + (−0.311 − 0.199i)17-s + ⋯ |
| L(s) = 1 | + (0.888 − 0.260i)2-s + (−0.222 − 1.54i)3-s + (−0.119 + 0.0769i)4-s + (−0.603 + 1.32i)5-s + (−0.602 − 1.31i)6-s + (1.51 − 0.444i)7-s + (−0.692 + 0.799i)8-s + (−1.39 + 0.408i)9-s + (−0.191 + 1.33i)10-s + (−0.0187 + 0.0411i)11-s + (0.145 + 0.168i)12-s + (−0.174 − 0.201i)13-s + (1.23 − 0.790i)14-s + (2.18 + 0.641i)15-s + (−0.347 + 0.761i)16-s + (−0.0754 − 0.0484i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03746 - 0.488229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03746 - 0.488229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (6.69 - 4.71i)T \) |
| good | 2 | \( 1 + (-1.25 + 0.368i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (0.385 + 2.68i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (1.35 - 2.95i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-4.00 + 1.17i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.0622 - 0.136i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.628 + 0.725i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.311 + 0.199i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (7.59 + 2.23i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (-0.206 - 1.43i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 - 1.51T + 29T^{2} \) |
| 31 | \( 1 + (-3.08 + 3.55i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 + (-3.83 - 2.46i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.83 - 4.38i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (0.238 + 1.65i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (7.23 - 4.65i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.70 + 7.73i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.894 + 1.95i)T + (-39.9 + 46.1i)T^{2} \) |
| 71 | \( 1 + (4.93 - 3.16i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (2.85 + 6.25i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.57 - 1.81i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.453 + 0.993i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (1.38 - 9.66i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + 3.44T + 97T^{2} \) |
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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
−14.46070937811487658987529740899, −13.54789054490153400996464267886, −12.56059598772801062147187292941, −11.46575493218501640475110249957, −10.98146550774306678969710109770, −8.249373833075628810884944765294, −7.48424235127038617816482273364, −6.23563430910932089423410363419, −4.41776711923411605056584042687, −2.47088745914474995044291560311,
4.29818899617109570060674391276, 4.60379763576456772872851962927, 5.67445864899398209607739852561, 8.365419571247936892563543216126, 9.098220449274899024610609649281, 10.57164899101556893430434516267, 11.78653302604365965554216146202, 12.72293237358782757931488779058, 14.32287705789936308721679009780, 14.99135954775564335676736744668
