| L(s) = 1 | + (−0.959 + 1.11i)2-s + (1.54 + 2.92i)3-s + (−0.0247 − 0.164i)4-s + (−2.03 − 0.919i)5-s + (−4.74 − 1.08i)6-s + (0.161 − 2.64i)7-s + (−2.28 − 1.43i)8-s + (−4.46 + 6.55i)9-s + (2.98 − 1.39i)10-s + (−2.72 + 1.86i)11-s + (0.441 − 0.325i)12-s + (0.818 + 2.33i)13-s + (2.79 + 2.71i)14-s + (−0.460 − 7.37i)15-s + (4.11 − 1.26i)16-s + (0.613 + 0.267i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.788i)2-s + (0.892 + 1.68i)3-s + (−0.0123 − 0.0820i)4-s + (−0.911 − 0.411i)5-s + (−1.93 − 0.442i)6-s + (0.0610 − 0.998i)7-s + (−0.808 − 0.507i)8-s + (−1.48 + 2.18i)9-s + (0.943 − 0.439i)10-s + (−0.822 + 0.560i)11-s + (0.127 − 0.0940i)12-s + (0.226 + 0.648i)13-s + (0.745 + 0.725i)14-s + (−0.118 − 1.90i)15-s + (1.02 − 0.317i)16-s + (0.148 + 0.0648i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.120486 - 0.808116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.120486 - 0.808116i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.03 + 0.919i)T \) |
| 7 | \( 1 + (-0.161 + 2.64i)T \) |
| good | 2 | \( 1 + (0.959 - 1.11i)T + (-0.298 - 1.97i)T^{2} \) |
| 3 | \( 1 + (-1.54 - 2.92i)T + (-1.68 + 2.47i)T^{2} \) |
| 11 | \( 1 + (2.72 - 1.86i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-0.818 - 2.33i)T + (-10.1 + 8.10i)T^{2} \) |
| 17 | \( 1 + (-0.613 - 0.267i)T + (11.5 + 12.4i)T^{2} \) |
| 19 | \( 1 + (-2.66 - 4.60i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.449 - 1.03i)T + (-15.6 + 16.8i)T^{2} \) |
| 29 | \( 1 + (-6.41 - 5.11i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.62 - 0.937i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.46 + 1.97i)T + (-10.9 + 35.3i)T^{2} \) |
| 41 | \( 1 + (-1.69 + 0.386i)T + (36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.734 - 1.16i)T + (-18.6 + 38.7i)T^{2} \) |
| 47 | \( 1 + (1.74 + 1.49i)T + (7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-5.07 + 6.87i)T + (-15.6 - 50.6i)T^{2} \) |
| 59 | \( 1 + (-0.829 - 0.769i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.862 + 5.72i)T + (-58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-10.0 + 2.70i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-6.40 - 8.03i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-1.62 + 1.40i)T + (10.8 - 72.1i)T^{2} \) |
| 79 | \( 1 + (1.66 - 0.963i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.63 + 1.97i)T + (64.8 + 51.7i)T^{2} \) |
| 89 | \( 1 + (12.6 + 8.62i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-9.12 + 9.12i)T - 97iT^{2} \) |
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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
−12.61821213809352068593373087148, −11.33159034607449063070143766233, −10.27265496013041648634335107271, −9.636506419418069129777982591615, −8.551904189940789704257999200033, −8.034905035433530393027809408826, −7.13316583060284036443491572718, −5.14923237338753192162273492487, −4.09594067001985122673871273263, −3.28690115999114387963318584769,
0.73904801953281429266136157123, 2.56535376547626325004751813384, 2.98695091128966641833563197390, 5.68228872473374183534690554228, 6.80007821845223356694794434007, 8.104129690209303307590779088121, 8.343173442696407325831500613598, 9.419077866471488608987902420269, 10.84341415418000563886535037983, 11.74892315125840508610696037454
