| 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 | \( 1 + (-2.39 + 3.30i)T + (-29.9 - 92.2i)T^{2} \) |
| 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
−14.52419888048096808115743750265, −13.36452240975580352634533665426, −11.58358807259694959428925233838, −10.89426514216705909398271297887, −10.19080453367031794446632751073, −9.060791157057220885071641268426, −8.494036536765143194785649328791, −5.47031189448311178682347473744, −4.06036650935908505488938668861, −2.58577087516365328455981367095,
1.95085801287966209669571111213, 5.49827744854509406603333664099, 6.68258948662717421350528230333, 7.57482651843062832979047518432, 8.247783496370522501644818493345, 9.471864905507487225657907981155, 11.45768182805862419806504208805, 12.66766272646626185351873069205, 13.72423475197242368160933015513, 14.53818353654084814338369556590
