| L(s) = 1 | + (−1.08 + 0.912i)2-s + (0.332 − 1.97i)4-s + (−0.114 + 0.136i)5-s + (2.42 + 0.881i)7-s + (1.44 + 2.43i)8-s + (−0.000918 − 0.252i)10-s + (0.599 + 0.714i)11-s + (−0.368 − 0.0649i)13-s + (−3.41 + 1.25i)14-s + (−3.77 − 1.31i)16-s + (2.26 − 3.91i)17-s + (1.97 − 1.14i)19-s + (0.231 + 0.271i)20-s + (−1.29 − 0.224i)22-s + (−4.40 + 1.60i)23-s + ⋯ |
| L(s) = 1 | + (−0.763 + 0.645i)2-s + (0.166 − 0.986i)4-s + (−0.0513 + 0.0611i)5-s + (0.914 + 0.333i)7-s + (0.509 + 0.860i)8-s + (−0.000290 − 0.0798i)10-s + (0.180 + 0.215i)11-s + (−0.102 − 0.0180i)13-s + (−0.913 + 0.336i)14-s + (−0.944 − 0.328i)16-s + (0.548 − 0.950i)17-s + (0.453 − 0.261i)19-s + (0.0517 + 0.0607i)20-s + (−0.277 − 0.0478i)22-s + (−0.918 + 0.334i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04389 + 0.479245i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04389 + 0.479245i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.08 - 0.912i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.114 - 0.136i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.42 - 0.881i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.599 - 0.714i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.368 + 0.0649i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.26 + 3.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.97 + 1.14i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.40 - 1.60i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 0.279i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.17 + 2.97i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.82 - 2.20i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.238 - 1.35i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.28 - 3.91i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-7.70 - 2.80i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 9.07iT - 53T^{2} \) |
| 59 | \( 1 + (-9.44 + 11.2i)T + (-10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (5.07 - 13.9i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.72 + 0.303i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.77 - 4.80i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.77 + 8.26i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.704 + 3.99i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-15.0 + 2.65i)T + (77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (8.16 + 14.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.94 - 7.50i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.47672856448578704174853253905, −9.643506533111774154056736861389, −8.941432829057483186554705990993, −7.88715797641660154490440749530, −7.45872508705054170901119761123, −6.27500879371853744690175834172, −5.35168222118304981870202409095, −4.48744960772721706298859607748, −2.62809071546064717889267437375, −1.20306549528001572665875268538,
1.03983713337184133683233609033, 2.31950020117867116298442587976, 3.71566550916061408401383255541, 4.60615368731312853070325602010, 6.04360682844831479075230169873, 7.19288533334434714176845554538, 8.182063614965336909555102983154, 8.486833423026267998071610713365, 9.792563225171138922852746938538, 10.39232416123919349431939269677
