| L(s) = 1 | + (1.25 − 0.368i)2-s + (0.0280 + 0.195i)3-s + (−0.239 + 0.153i)4-s + (0.681 − 1.49i)5-s + (0.107 + 0.235i)6-s + (−3.00 + 0.881i)7-s + (−1.95 + 2.26i)8-s + (2.84 − 0.834i)9-s + (0.305 − 2.12i)10-s + (−0.886 + 1.94i)11-s + (−0.0367 − 0.0424i)12-s + (−3.83 − 4.42i)13-s + (−3.44 + 2.21i)14-s + (0.310 + 0.0912i)15-s + (−1.39 + 3.04i)16-s + (1.43 + 0.924i)17-s + ⋯ |
| L(s) = 1 | + (0.888 − 0.260i)2-s + (0.0162 + 0.112i)3-s + (−0.119 + 0.0769i)4-s + (0.304 − 0.667i)5-s + (0.0438 + 0.0959i)6-s + (−1.13 + 0.333i)7-s + (−0.692 + 0.799i)8-s + (0.947 − 0.278i)9-s + (0.0967 − 0.673i)10-s + (−0.267 + 0.585i)11-s + (−0.0106 − 0.0122i)12-s + (−1.06 − 1.22i)13-s + (−0.921 + 0.592i)14-s + (0.0802 + 0.0235i)15-s + (−0.347 + 0.761i)16-s + (0.348 + 0.224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19181 - 0.123365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19181 - 0.123365i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (2.29 - 7.85i)T \) |
| good | 2 | \( 1 + (-1.25 + 0.368i)T + (1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (-0.0280 - 0.195i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (-0.681 + 1.49i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (3.00 - 0.881i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.886 - 1.94i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (3.83 + 4.42i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.43 - 0.924i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-4.99 - 1.46i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (0.305 + 2.12i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 + 3.38T + 29T^{2} \) |
| 31 | \( 1 + (-4.27 + 4.93i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + (1.44 + 0.930i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (7.43 + 4.77i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-1.31 - 9.13i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (7.51 - 4.83i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (2.14 - 2.47i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.10 + 2.42i)T + (-39.9 + 46.1i)T^{2} \) |
| 71 | \( 1 + (11.1 - 7.16i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (5.26 + 11.5i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.65 - 1.90i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.55 + 5.60i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (1.26 - 8.82i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 - 8.00T + 97T^{2} \) |
| 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
−14.76730694266312965905233622402, −13.24414662849307703843684324251, −12.76154743098707378521551966955, −12.09156173430593919395515901529, −10.04857135346851063490101973925, −9.345491426528337955854289018627, −7.64752399089256093127030412916, −5.81930359738603782285552685616, −4.66251074837656079124816551064, −3.06195137789712560775571887286,
3.23241870794106484310981839179, 4.81843081960873247110985550496, 6.39516401495845659524562709954, 7.21163073080651790327813893344, 9.499263907405420367490238389304, 10.10265101541870985312460957842, 11.84830037612418384859573961065, 13.06295304556786711145699376191, 13.73468191973516581255264849764, 14.59421260308333939363868435139
