| L(s) = 1 | + (0.370 − 0.928i)2-s + (0.994 + 0.108i)3-s + (−0.725 − 0.687i)4-s + (0.988 − 1.16i)5-s + (0.468 − 0.883i)6-s + (−2.12 − 1.28i)7-s + (−0.907 + 0.419i)8-s + (0.976 + 0.214i)9-s + (−0.715 − 1.34i)10-s + (0.211 − 3.89i)11-s + (−0.647 − 0.762i)12-s + (1.49 − 0.329i)13-s + (−1.97 + 1.50i)14-s + (1.10 − 1.05i)15-s + (0.0541 + 0.998i)16-s + (1.42 − 0.856i)17-s + ⋯ |
| L(s) = 1 | + (0.261 − 0.656i)2-s + (0.573 + 0.0624i)3-s + (−0.362 − 0.343i)4-s + (0.442 − 0.520i)5-s + (0.191 − 0.360i)6-s + (−0.804 − 0.484i)7-s + (−0.320 + 0.148i)8-s + (0.325 + 0.0716i)9-s + (−0.226 − 0.426i)10-s + (0.0636 − 1.17i)11-s + (−0.186 − 0.220i)12-s + (0.415 − 0.0913i)13-s + (−0.528 + 0.401i)14-s + (0.286 − 0.271i)15-s + (0.0135 + 0.249i)16-s + (0.345 − 0.207i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0968 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17139 - 1.29089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17139 - 1.29089i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.370 + 0.928i)T \) |
| 3 | \( 1 + (-0.994 - 0.108i)T \) |
| 59 | \( 1 + (7.64 - 0.750i)T \) |
| good | 5 | \( 1 + (-0.988 + 1.16i)T + (-0.808 - 4.93i)T^{2} \) |
| 7 | \( 1 + (2.12 + 1.28i)T + (3.27 + 6.18i)T^{2} \) |
| 11 | \( 1 + (-0.211 + 3.89i)T + (-10.9 - 1.18i)T^{2} \) |
| 13 | \( 1 + (-1.49 + 0.329i)T + (11.7 - 5.45i)T^{2} \) |
| 17 | \( 1 + (-1.42 + 0.856i)T + (7.96 - 15.0i)T^{2} \) |
| 19 | \( 1 + (-0.139 + 0.503i)T + (-16.2 - 9.79i)T^{2} \) |
| 23 | \( 1 + (-1.78 - 2.63i)T + (-8.51 + 21.3i)T^{2} \) |
| 29 | \( 1 + (-3.64 - 9.15i)T + (-21.0 + 19.9i)T^{2} \) |
| 31 | \( 1 + (1.84 + 6.65i)T + (-26.5 + 15.9i)T^{2} \) |
| 37 | \( 1 + (3.18 + 1.47i)T + (23.9 + 28.1i)T^{2} \) |
| 41 | \( 1 + (1.26 - 1.86i)T + (-15.1 - 38.0i)T^{2} \) |
| 43 | \( 1 + (0.159 + 2.93i)T + (-42.7 + 4.64i)T^{2} \) |
| 47 | \( 1 + (-8.34 - 9.82i)T + (-7.60 + 46.3i)T^{2} \) |
| 53 | \( 1 + (1.41 - 2.67i)T + (-29.7 - 43.8i)T^{2} \) |
| 61 | \( 1 + (4.37 - 10.9i)T + (-44.2 - 41.9i)T^{2} \) |
| 67 | \( 1 + (3.00 - 1.39i)T + (43.3 - 51.0i)T^{2} \) |
| 71 | \( 1 + (-4.71 - 5.55i)T + (-11.4 + 70.0i)T^{2} \) |
| 73 | \( 1 + (-3.51 + 2.67i)T + (19.5 - 70.3i)T^{2} \) |
| 79 | \( 1 + (-4.08 + 0.443i)T + (77.1 - 16.9i)T^{2} \) |
| 83 | \( 1 + (-15.0 + 5.06i)T + (66.0 - 50.2i)T^{2} \) |
| 89 | \( 1 + (-1.07 - 2.70i)T + (-64.6 + 61.2i)T^{2} \) |
| 97 | \( 1 + (-3.51 - 2.66i)T + (25.9 + 93.4i)T^{2} \) |
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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
−11.09724197450892433371941571363, −10.36696195147618127256266281748, −9.282522254963639637838783252545, −8.856063383668619400425446551147, −7.52907623123952338208533611355, −6.21413002557588460904085108544, −5.19309762508851335052591065281, −3.76991856909058613381940065964, −2.97230468220564566980518545920, −1.17548276910235360263455866250,
2.30547808821865463960324877654, 3.50867384938702188251166483611, 4.81769469909930995782401074272, 6.20026267663465416158916866212, 6.77461804740616299557452894863, 7.87310647051130548660849906386, 8.904398822265950362504339599719, 9.748915698806507968907747450973, 10.49184251483054427492758163487, 12.11606781427173465375742530236
