| L(s) = 1 | + (−0.842 + 1.89i)2-s + (0.653 − 3.07i)3-s + (−1.52 − 1.69i)4-s + (1.18 + 0.124i)5-s + (5.26 + 3.82i)6-s + (2.63 − 0.192i)7-s + (0.563 − 0.183i)8-s + (−6.27 − 2.79i)9-s + (−1.22 + 2.13i)10-s + (−0.746 + 3.23i)11-s + (−6.22 + 3.59i)12-s + (−0.000347 + 0.000252i)13-s + (−1.85 + 5.15i)14-s + (1.15 − 3.54i)15-s + (0.349 − 3.32i)16-s + (−1.68 + 0.749i)17-s + ⋯ |
| L(s) = 1 | + (−0.595 + 1.33i)2-s + (0.377 − 1.77i)3-s + (−0.764 − 0.849i)4-s + (0.528 + 0.0555i)5-s + (2.14 + 1.56i)6-s + (0.997 − 0.0727i)7-s + (0.199 − 0.0647i)8-s + (−2.09 − 0.931i)9-s + (−0.388 + 0.673i)10-s + (−0.225 + 0.974i)11-s + (−1.79 + 1.03i)12-s + (−9.63e−5 + 6.99e−5i)13-s + (−0.496 + 1.37i)14-s + (0.297 − 0.916i)15-s + (0.0874 − 0.831i)16-s + (−0.408 + 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.833043 + 0.105233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.833043 + 0.105233i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.63 + 0.192i)T \) |
| 11 | \( 1 + (0.746 - 3.23i)T \) |
| good | 2 | \( 1 + (0.842 - 1.89i)T + (-1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.653 + 3.07i)T + (-2.74 - 1.22i)T^{2} \) |
| 5 | \( 1 + (-1.18 - 0.124i)T + (4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (0.000347 - 0.000252i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.68 - 0.749i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (1.29 - 1.43i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.83 - 4.91i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.88 + 1.58i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.431 + 0.0453i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.367 - 0.0782i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (0.760 + 2.34i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 5.03iT - 43T^{2} \) |
| 47 | \( 1 + (-1.37 - 1.24i)T + (4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (0.138 + 1.31i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-9.06 + 8.16i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.795 + 7.56i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-6.72 + 11.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.82 + 4.96i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-9.12 - 10.1i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (0.667 - 1.49i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (7.86 + 5.71i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (6.24 - 3.60i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.39 - 3.30i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−14.53818353654084814338369556590, −13.72423475197242368160933015513, −12.66766272646626185351873069205, −11.45768182805862419806504208805, −9.471864905507487225657907981155, −8.247783496370522501644818493345, −7.57482651843062832979047518432, −6.68258948662717421350528230333, −5.49827744854509406603333664099, −1.95085801287966209669571111213,
2.58577087516365328455981367095, 4.06036650935908505488938668861, 5.47031189448311178682347473744, 8.494036536765143194785649328791, 9.060791157057220885071641268426, 10.19080453367031794446632751073, 10.89426514216705909398271297887, 11.58358807259694959428925233838, 13.36452240975580352634533665426, 14.52419888048096808115743750265
