| L(s) = 1 | − 2.34i·2-s − 5.73i·3-s − 1.51·4-s − 13.4·6-s + 9.56·7-s − 5.83i·8-s − 23.9·9-s + 4.15·11-s + 8.68i·12-s − 4.55i·13-s − 22.4i·14-s − 19.7·16-s − 27.9·17-s + 56.1i·18-s + (7.04 + 17.6i)19-s + ⋯ |
| L(s) = 1 | − 1.17i·2-s − 1.91i·3-s − 0.378·4-s − 2.24·6-s + 1.36·7-s − 0.729i·8-s − 2.65·9-s + 0.377·11-s + 0.723i·12-s − 0.350i·13-s − 1.60i·14-s − 1.23·16-s − 1.64·17-s + 3.11i·18-s + (0.370 + 0.928i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.734205580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734205580\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (-7.04 - 17.6i)T \) |
| good | 2 | \( 1 + 2.34iT - 4T^{2} \) |
| 3 | \( 1 + 5.73iT - 9T^{2} \) |
| 7 | \( 1 - 9.56T + 49T^{2} \) |
| 11 | \( 1 - 4.15T + 121T^{2} \) |
| 13 | \( 1 + 4.55iT - 169T^{2} \) |
| 17 | \( 1 + 27.9T + 289T^{2} \) |
| 23 | \( 1 + 13.9T + 529T^{2} \) |
| 29 | \( 1 - 14.6iT - 841T^{2} \) |
| 31 | \( 1 + 28.9iT - 961T^{2} \) |
| 37 | \( 1 + 30.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 44.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 69.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 30.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 54.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 31.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 99.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 3.05iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 23.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 21.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 77.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 93.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 126. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 77.2iT - 9.40e3T^{2} \) |
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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
−10.74549596435960875947441152696, −9.186543449232642162994180766736, −8.272719548671794634597197036207, −7.48394186819072605497611000730, −6.62503033819965583920435715014, −5.54665525879931856958718988963, −3.95915974472213109656538806324, −2.34803982924418696537807792395, −1.83723809005254032651728135305, −0.69133985938167676438258758575,
2.49710897983064565252455433320, 4.27387216914300714044865180817, 4.71846017529724996713557326955, 5.60905104322476061240444629168, 6.68633333923405850482192193972, 7.985209005630247655822425084965, 8.754724882220095301655115169150, 9.310592413095499127361390568850, 10.59867258232692916548566936032, 11.25571109099286420166874680775
