L(s) = 1 | + (−0.270 + 1.38i)2-s + (−1.73 − 0.0414i)3-s + (−1.85 − 0.749i)4-s + (−0.0583 + 0.160i)5-s + (0.525 − 2.39i)6-s + (−0.984 − 0.173i)7-s + (1.54 − 2.37i)8-s + (2.99 + 0.143i)9-s + (−0.206 − 0.124i)10-s + (−1.14 + 0.416i)11-s + (3.17 + 1.37i)12-s + (2.79 − 2.34i)13-s + (0.507 − 1.32i)14-s + (0.107 − 0.275i)15-s + (2.87 + 2.78i)16-s + (−0.976 + 0.563i)17-s + ⋯ |
L(s) = 1 | + (−0.191 + 0.981i)2-s + (−0.999 − 0.0239i)3-s + (−0.927 − 0.374i)4-s + (−0.0261 + 0.0717i)5-s + (0.214 − 0.976i)6-s + (−0.372 − 0.0656i)7-s + (0.545 − 0.838i)8-s + (0.998 + 0.0477i)9-s + (−0.0654 − 0.0393i)10-s + (−0.344 + 0.125i)11-s + (0.917 + 0.397i)12-s + (0.776 − 0.651i)13-s + (0.135 − 0.352i)14-s + (0.0278 − 0.0710i)15-s + (0.718 + 0.695i)16-s + (−0.236 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.281289 + 0.610651i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.281289 + 0.610651i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.270 - 1.38i)T \) |
| 3 | \( 1 + (1.73 + 0.0414i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
good | 5 | \( 1 + (0.0583 - 0.160i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (1.14 - 0.416i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.79 + 2.34i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.976 - 0.563i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.62 - 0.936i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.245 - 1.39i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.77 - 3.30i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.93 - 0.516i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.20 + 2.09i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.85 - 2.21i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.37 - 6.52i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.989 - 5.61i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 + (3.41 + 1.24i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.338 + 1.92i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.29 + 6.30i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.33 - 12.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.29 - 3.98i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.27 - 3.89i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.41 - 1.18i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-12.6 - 7.32i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.86 + 2.50i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.65527350799973594328439551080, −9.741256755583650663325607088132, −8.951021084606077670471930352724, −7.78758057520136452783246596346, −7.14165173386080172867964786677, −6.17797915663561549268271938230, −5.56774789384325887754779117089, −4.64674954138424784021972409744, −3.47719605459983317096048357565, −1.15588387631105307893873145279,
0.50748401765839711662650296601, 2.00317784128403612567503230839, 3.48854527981671074799669056937, 4.46560569989227405849497876228, 5.38705959598133402223211300671, 6.41559990919981116244164337612, 7.43959069626170667007780208776, 8.624524302210356820014351531812, 9.369724665066910348255820087497, 10.29676193221875596173475904977