| L(s) = 1 | + (−0.453 + 0.891i)2-s + (0.865 + 0.562i)3-s + (−0.587 − 0.809i)4-s + (−0.633 − 2.14i)5-s + (−0.893 + 0.516i)6-s + (−0.913 − 2.37i)7-s + (0.987 − 0.156i)8-s + (−0.786 − 1.76i)9-s + (2.19 + 0.409i)10-s + (1.21 − 0.127i)11-s + (−0.0540 − 1.03i)12-s + (0.164 − 3.12i)13-s + (2.53 + 0.266i)14-s + (0.657 − 2.21i)15-s + (−0.309 + 0.951i)16-s + (−1.30 − 1.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 + 0.630i)2-s + (0.499 + 0.324i)3-s + (−0.293 − 0.404i)4-s + (−0.283 − 0.959i)5-s + (−0.364 + 0.210i)6-s + (−0.345 − 0.898i)7-s + (0.349 − 0.0553i)8-s + (−0.262 − 0.589i)9-s + (0.695 + 0.129i)10-s + (0.365 − 0.0384i)11-s + (−0.0155 − 0.297i)12-s + (0.0454 − 0.868i)13-s + (0.677 + 0.0711i)14-s + (0.169 − 0.571i)15-s + (−0.0772 + 0.237i)16-s + (−0.315 − 0.389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.989062 - 0.376117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.989062 - 0.376117i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.453 - 0.891i)T \) |
| 5 | \( 1 + (0.633 + 2.14i)T \) |
| 31 | \( 1 + (-5.43 + 1.19i)T \) |
| good | 3 | \( 1 + (-0.865 - 0.562i)T + (1.22 + 2.74i)T^{2} \) |
| 7 | \( 1 + (0.913 + 2.37i)T + (-5.20 + 4.68i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 0.127i)T + (10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-0.164 + 3.12i)T + (-12.9 - 1.35i)T^{2} \) |
| 17 | \( 1 + (1.30 + 1.60i)T + (-3.53 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-1.02 - 0.926i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.488 - 3.08i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.86 - 5.75i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (1.30 + 0.348i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-6.62 - 1.40i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (5.33 - 0.279i)T + (42.7 - 4.49i)T^{2} \) |
| 47 | \( 1 + (-5.82 + 2.96i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (2.94 + 1.13i)T + (39.3 + 35.4i)T^{2} \) |
| 59 | \( 1 + (-1.93 - 9.12i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 10.7iT - 61T^{2} \) |
| 67 | \( 1 + (-0.419 + 0.112i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-4.32 + 1.92i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (8.51 - 10.5i)T + (-15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-0.919 + 8.74i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.842 - 1.29i)T + (-33.7 + 75.8i)T^{2} \) |
| 89 | \( 1 + (8.14 - 5.91i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (16.5 + 2.62i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.58252280175443002326657822650, −10.31076296932221374464762295278, −9.498143146774245498896083364538, −8.735004176298775550769096772571, −7.892355701885740105629745783603, −6.86680374975330770298880010748, −5.63528409214257656850430911515, −4.42349315506013015506896469978, −3.37865981860248589704568395840, −0.841408082703780381383509113893,
2.16202250185857891147437567451, 2.96312562799559845253239078904, 4.36197846372620442124803000681, 6.07276845909771772989438968526, 7.09332206060393957285183395946, 8.189655010405060082039788704172, 8.949619869721108167826668532858, 9.948481495877899684310250510837, 10.96688091694011577472629789105, 11.69327594659919821949399822113
