| L(s) = 1 | + 1.40·2-s + 2.91·3-s − 2.02·4-s + 4.10·6-s − 11.7i·7-s − 8.46·8-s − 0.493·9-s − 6.36·11-s − 5.89·12-s − 17.7·13-s − 16.4i·14-s − 3.82·16-s − 17.5i·17-s − 0.694·18-s + (16.8 − 8.86i)19-s + ⋯ |
| L(s) = 1 | + 0.703·2-s + 0.972·3-s − 0.505·4-s + 0.683·6-s − 1.67i·7-s − 1.05·8-s − 0.0548·9-s − 0.578·11-s − 0.491·12-s − 1.36·13-s − 1.17i·14-s − 0.238·16-s − 1.03i·17-s − 0.0385·18-s + (0.884 − 0.466i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.671935203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671935203\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (-16.8 + 8.86i)T \) |
| good | 2 | \( 1 - 1.40T + 4T^{2} \) |
| 3 | \( 1 - 2.91T + 9T^{2} \) |
| 7 | \( 1 + 11.7iT - 49T^{2} \) |
| 11 | \( 1 + 6.36T + 121T^{2} \) |
| 13 | \( 1 + 17.7T + 169T^{2} \) |
| 17 | \( 1 + 17.5iT - 289T^{2} \) |
| 23 | \( 1 + 1.53iT - 529T^{2} \) |
| 29 | \( 1 - 39.9iT - 841T^{2} \) |
| 31 | \( 1 - 9.24iT - 961T^{2} \) |
| 37 | \( 1 - 56.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 18.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 61.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 66.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 9.42iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2.55T + 4.48e3T^{2} \) |
| 71 | \( 1 + 53.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 115. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 95.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 143. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 19.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 163.T + 9.40e3T^{2} \) |
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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
−10.29686105952846107999205641870, −9.583204318399543490788207326519, −8.745240595928002179034650243199, −7.56662522890338994283296261738, −7.10535351351403748665245204993, −5.36880405409823761289684911642, −4.59856544433538427188815807127, −3.53128842191678917688891537736, −2.68302038998568924725645142838, −0.45961731345704716101741925328,
2.39283398731302130472365703502, 2.98033409199776861470918893887, 4.33306022323857130543244871368, 5.45736002182127425892083696853, 6.03567978241974686069855470957, 7.87000428002098330188700796748, 8.323417643291753007246467798247, 9.454044253803909234873566285900, 9.689203968778364132695554463399, 11.44654798390808730086230994536
