| L(s) = 1 | + 2.82i·2-s + (2.39 − 8.67i)3-s − 8.00·4-s + 4.41i·5-s + (24.5 + 6.76i)6-s − 13.1·7-s − 22.6i·8-s + (−69.5 − 41.5i)9-s − 12.4·10-s + 230. i·11-s + (−19.1 + 69.4i)12-s + 86.7·13-s − 37.0i·14-s + (38.2 + 10.5i)15-s + 64.0·16-s − 170. i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.265 − 0.963i)3-s − 0.500·4-s + 0.176i·5-s + (0.681 + 0.188i)6-s − 0.267·7-s − 0.353i·8-s + (−0.858 − 0.512i)9-s − 0.124·10-s + 1.90i·11-s + (−0.132 + 0.481i)12-s + 0.513·13-s − 0.189i·14-s + (0.170 + 0.0469i)15-s + 0.250·16-s − 0.589i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.963i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.265 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.641022667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.641022667\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 + (-2.39 + 8.67i)T \) |
| 29 | \( 1 + 156. iT \) |
| good | 5 | \( 1 - 4.41iT - 625T^{2} \) |
| 7 | \( 1 + 13.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 230. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 86.7T + 2.85e4T^{2} \) |
| 17 | \( 1 + 170. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 397.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 986. iT - 2.79e5T^{2} \) |
| 31 | \( 1 - 1.36e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 457.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.06e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 466.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 664. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.12e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.69e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 3.68e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.22e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 8.22e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 568.T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.58e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.17e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.31e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 2.56e3T + 8.85e7T^{2} \) |
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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
−12.41614825957409955505156005406, −11.56943187638362314519118285405, −9.899440023892314138303396175066, −9.130572070627174418131633138871, −7.71014400195229391042511757194, −7.20467064498567102249908388985, −6.15907251714612235472471818016, −4.80502410445723501980743869818, −3.06343919912375602875893018064, −1.35111219970245837046140836778,
0.66627916583983977361419479606, 2.85650912281525679713694312572, 3.72833616369319144661095348186, 5.06875952756731591721406524729, 6.23151309749773151663641611999, 8.397570439448860579901139987258, 8.729556845371263513881366966617, 10.02344097477814753674104097874, 10.80366240269634512005535755588, 11.54711232068267421393015511222
