| L(s) = 1 | + (0.877 + 0.479i)2-s + (0.997 + 0.0713i)3-s + (0.540 + 0.841i)4-s + (−0.516 − 2.17i)5-s + (0.841 + 0.540i)6-s + (−1.41 − 3.79i)7-s + (0.0713 + 0.997i)8-s + (0.989 + 0.142i)9-s + (0.589 − 2.15i)10-s + (0.380 − 1.29i)11-s + (0.479 + 0.877i)12-s + (−3.10 − 1.15i)13-s + (0.576 − 4.01i)14-s + (−0.359 − 2.20i)15-s + (−0.415 + 0.909i)16-s + (1.22 − 0.265i)17-s + ⋯ |
| L(s) = 1 | + (0.620 + 0.338i)2-s + (0.575 + 0.0411i)3-s + (0.270 + 0.420i)4-s + (−0.230 − 0.972i)5-s + (0.343 + 0.220i)6-s + (−0.535 − 1.43i)7-s + (0.0252 + 0.352i)8-s + (0.329 + 0.0474i)9-s + (0.186 − 0.682i)10-s + (0.114 − 0.390i)11-s + (0.138 + 0.253i)12-s + (−0.861 − 0.321i)13-s + (0.154 − 1.07i)14-s + (−0.0929 − 0.569i)15-s + (−0.103 + 0.227i)16-s + (0.296 − 0.0644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.03262 - 0.983328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03262 - 0.983328i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.877 - 0.479i)T \) |
| 3 | \( 1 + (-0.997 - 0.0713i)T \) |
| 5 | \( 1 + (0.516 + 2.17i)T \) |
| 23 | \( 1 + (1.15 - 4.65i)T \) |
| good | 7 | \( 1 + (1.41 + 3.79i)T + (-5.29 + 4.58i)T^{2} \) |
| 11 | \( 1 + (-0.380 + 1.29i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (3.10 + 1.15i)T + (9.82 + 8.51i)T^{2} \) |
| 17 | \( 1 + (-1.22 + 0.265i)T + (15.4 - 7.06i)T^{2} \) |
| 19 | \( 1 + (-6.03 + 3.87i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-5.50 + 8.56i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (4.64 - 5.35i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-4.71 - 6.30i)T + (-10.4 + 35.5i)T^{2} \) |
| 41 | \( 1 + (0.384 + 2.67i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.0327 + 0.458i)T + (-42.5 - 6.11i)T^{2} \) |
| 47 | \( 1 + (-4.63 - 4.63i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.204 - 0.0762i)T + (40.0 - 34.7i)T^{2} \) |
| 59 | \( 1 + (-7.48 + 3.41i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-8.31 - 7.20i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (5.54 - 10.1i)T + (-36.2 - 56.3i)T^{2} \) |
| 71 | \( 1 + (3.52 - 1.03i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (2.22 - 10.2i)T + (-66.4 - 30.3i)T^{2} \) |
| 79 | \( 1 + (4.99 + 10.9i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-5.17 + 3.87i)T + (23.3 - 79.6i)T^{2} \) |
| 89 | \( 1 + (-8.78 - 10.1i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.05 + 1.53i)T + (27.3 + 93.0i)T^{2} \) |
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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
−10.09725830424007542472571889926, −9.562907901574256989302186473666, −8.450024337961334327072271881299, −7.53517395945807703960612166834, −7.09578650222585235749431476013, −5.69250265942015954464747483253, −4.70868437615696567548025510527, −3.88854577584950766089711205410, −2.93547930248763283822671599146, −0.949909426000244488953428392794,
2.14262840663412036642395886846, 2.87397677166178453233080282672, 3.78508809167220198374279212956, 5.14369995630030917325240007196, 6.09247780289214194195577940064, 7.00459038329940986882028024309, 7.86757863170720485283163207031, 9.100648436474544584897266849237, 9.762006673346501185926985258376, 10.54096340827926596258775970023
