| L(s) = 1 | + (1.40 + 0.125i)2-s + (0.654 − 0.755i)3-s + (1.96 + 0.353i)4-s + (0.329 − 2.29i)5-s + (1.01 − 0.982i)6-s + (0.102 + 0.223i)7-s + (2.72 + 0.745i)8-s + (−0.142 − 0.989i)9-s + (0.752 − 3.18i)10-s + (−1.07 − 3.67i)11-s + (1.55 − 1.25i)12-s + (−4.96 − 2.26i)13-s + (0.115 + 0.327i)14-s + (−1.51 − 1.75i)15-s + (3.74 + 1.39i)16-s + (3.16 + 4.92i)17-s + ⋯ |
| L(s) = 1 | + (0.996 + 0.0888i)2-s + (0.378 − 0.436i)3-s + (0.984 + 0.176i)4-s + (0.147 − 1.02i)5-s + (0.415 − 0.401i)6-s + (0.0385 + 0.0844i)7-s + (0.964 + 0.263i)8-s + (−0.0474 − 0.329i)9-s + (0.237 − 1.00i)10-s + (−0.325 − 1.10i)11-s + (0.449 − 0.362i)12-s + (−1.37 − 0.629i)13-s + (0.0309 + 0.0875i)14-s + (−0.391 − 0.451i)15-s + (0.937 + 0.348i)16-s + (0.767 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.72723 - 1.23205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.72723 - 1.23205i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.125i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (0.134 - 4.79i)T \) |
| good | 5 | \( 1 + (-0.329 + 2.29i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.102 - 0.223i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (1.07 + 3.67i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (4.96 + 2.26i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.16 - 4.92i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (2.56 - 3.98i)T + (-7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.43 - 8.46i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.84 + 2.46i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.0873 + 0.607i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.08 + 7.51i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (4.07 + 3.53i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 7.48iT - 47T^{2} \) |
| 53 | \( 1 + (3.61 + 7.92i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (2.19 - 4.80i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-2.00 - 2.31i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.858 + 2.92i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (4.30 - 14.6i)T + (-59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.82 - 5.66i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (2.27 - 4.97i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.0564 + 0.00811i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (12.4 + 10.8i)T + (12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (15.6 + 2.25i)T + (93.0 + 27.3i)T^{2} \) |
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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
−10.75382073653253392171656710211, −9.923769846851734058863831827006, −8.505641354107571912051074970290, −8.064544852312630560286710848333, −6.98572506377086776834247582447, −5.71683002346990935385090753087, −5.26881731134041401160721675948, −3.92190488889674635284809803133, −2.84640043318234574599144842002, −1.42917136162576944233632625817,
2.42357257526597509649429714387, 2.83926467716461638985737581774, 4.48888021641195756802268171450, 4.87468007323700907965891103728, 6.46298564859873542394910450903, 7.07317445783832643000662715483, 7.912806424735727416804603580320, 9.592481081074585218849918895104, 10.08076167391339115928736091582, 10.90305146544705442649949268252
