| L(s) = 1 | − 2.07i·2-s + 11.7·4-s + 48.7i·5-s − 18.1·7-s − 57.4i·8-s + 101.·10-s − 148. i·11-s + 163.·13-s + 37.6i·14-s + 68.1·16-s − 244. i·17-s + 48.5·19-s + 570. i·20-s − 308.·22-s − 859. i·23-s + ⋯ |
| L(s) = 1 | − 0.518i·2-s + 0.731·4-s + 1.94i·5-s − 0.370·7-s − 0.897i·8-s + 1.01·10-s − 1.22i·11-s + 0.966·13-s + 0.192i·14-s + 0.266·16-s − 0.845i·17-s + 0.134·19-s + 1.42i·20-s − 0.636·22-s − 1.62i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.619187364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.619187364\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 453. iT \) |
| good | 2 | \( 1 + 2.07iT - 16T^{2} \) |
| 5 | \( 1 - 48.7iT - 625T^{2} \) |
| 7 | \( 1 + 18.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 148. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 163.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 244. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 48.5T + 1.30e5T^{2} \) |
| 23 | \( 1 + 859. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 199. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.34e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.44e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.06e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.17e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.41e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.94e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 - 6.27e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.90e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 924. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.74e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 7.23e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 2.64e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.03e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.82e3T + 8.85e7T^{2} \) |
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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
−10.47535191774389561224587131098, −9.636948971422690630032450961780, −8.272976695177029713413844077826, −7.22936138826494561462814011912, −6.41629094311962366385126415224, −6.02383649043148635616014887875, −3.91236630202793663916364752648, −2.97397559469608786361409737972, −2.54185886461875071939545053773, −0.76135647967693572446737526957,
1.08621488585625815243103930633, 1.97682109609034213280032266004, 3.78064609802173673887404622254, 4.86434812613369837013739896339, 5.70677384762501372842312938280, 6.58781595349128792419097768604, 7.84709445845340124121055725106, 8.312686088273172639467268633030, 9.382319407813353083615886342284, 10.07375124040673022112258829966
