| L(s) = 1 | + (0.667 − 1.24i)2-s + (−0.819 + 0.573i)3-s + (−1.10 − 1.66i)4-s + (3.18 − 0.278i)5-s + (0.168 + 1.40i)6-s + (−0.267 − 0.462i)7-s + (−2.81 + 0.270i)8-s + (0.342 − 0.939i)9-s + (1.77 − 4.15i)10-s + (5.43 − 1.45i)11-s + (1.86 + 0.728i)12-s + (−5.10 − 3.57i)13-s + (−0.754 + 0.0240i)14-s + (−2.44 + 2.05i)15-s + (−1.54 + 3.69i)16-s + (−0.691 + 0.251i)17-s + ⋯ |
| L(s) = 1 | + (0.472 − 0.881i)2-s + (−0.472 + 0.331i)3-s + (−0.554 − 0.832i)4-s + (1.42 − 0.124i)5-s + (0.0686 + 0.573i)6-s + (−0.100 − 0.174i)7-s + (−0.995 + 0.0954i)8-s + (0.114 − 0.313i)9-s + (0.562 − 1.31i)10-s + (1.63 − 0.438i)11-s + (0.537 + 0.210i)12-s + (−1.41 − 0.991i)13-s + (−0.201 + 0.00643i)14-s + (−0.631 + 0.530i)15-s + (−0.385 + 0.922i)16-s + (−0.167 + 0.0610i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15999 - 1.63187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15999 - 1.63187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.667 + 1.24i)T \) |
| 3 | \( 1 + (0.819 - 0.573i)T \) |
| 19 | \( 1 + (0.0676 - 4.35i)T \) |
| good | 5 | \( 1 + (-3.18 + 0.278i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (0.267 + 0.462i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.43 + 1.45i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (5.10 + 3.57i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.691 - 0.251i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.85 + 4.91i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-5.78 + 2.69i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-0.448 - 0.776i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.71 + 7.71i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.589 + 3.34i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.0274 + 0.314i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (2.15 - 5.91i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (9.60 + 0.840i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-2.73 + 5.87i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (0.196 + 0.0171i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-4.82 - 10.3i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (6.59 - 7.85i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (4.98 + 0.879i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.492 + 2.79i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.99 - 0.533i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.28 - 7.30i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.60 - 7.14i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.06668236280720688057155196980, −9.353993137356862079419665684200, −8.629409575453977987150325203297, −6.87217448401709981904310411903, −6.09141352942848619333823524833, −5.36989483853664224887348957939, −4.53263107225018768575409875751, −3.36612507281752866269239820173, −2.19713413374476475276251683784, −0.929901598144142863436053967193,
1.66586752892061052904303666216, 2.96969072258731608776664855771, 4.60271291888524593676800631154, 5.10315790118789825810413087774, 6.27901160323649527713938417343, 6.73890647284645343070697463797, 7.29521334034753135805063533350, 8.913341949364049558268878954567, 9.300974327424850280423420522229, 10.04344767810575809242222080456
