L(s) = 1 | + i·2-s + (2.36 − 1.36i)3-s − 4-s + (2.17 − 0.537i)5-s + (1.36 + 2.36i)6-s + (−2.77 − 1.60i)7-s − i·8-s + (2.21 − 3.83i)9-s + (0.537 + 2.17i)10-s − 0.912·11-s + (−2.36 + 1.36i)12-s + (4.63 + 2.67i)13-s + (1.60 − 2.77i)14-s + (4.39 − 4.22i)15-s + 16-s + (1.04 + 0.604i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (1.36 − 0.786i)3-s − 0.5·4-s + (0.970 − 0.240i)5-s + (0.556 + 0.963i)6-s + (−1.04 − 0.606i)7-s − 0.353i·8-s + (0.738 − 1.27i)9-s + (0.169 + 0.686i)10-s − 0.275·11-s + (−0.681 + 0.393i)12-s + (1.28 + 0.742i)13-s + (0.428 − 0.742i)14-s + (1.13 − 1.09i)15-s + 0.250·16-s + (0.253 + 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.15260 - 0.168120i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15260 - 0.168120i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-2.17 + 0.537i)T \) |
| 43 | \( 1 + (0.613 - 6.52i)T \) |
good | 3 | \( 1 + (-2.36 + 1.36i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (2.77 + 1.60i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 0.912T + 11T^{2} \) |
| 13 | \( 1 + (-4.63 - 2.67i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 0.604i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.17 + 2.04i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0863 - 0.0498i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.05 - 7.02i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.12 + 7.14i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0240 - 0.0139i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 0.174T + 41T^{2} \) |
| 47 | \( 1 - 7.23iT - 47T^{2} \) |
| 53 | \( 1 + (-1.00 + 0.582i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 1.44T + 59T^{2} \) |
| 61 | \( 1 + (3.43 - 5.95i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (14.0 - 8.09i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.176 - 0.306i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.0 - 5.77i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.17 - 12.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.42 + 5.44i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.18 + 3.78i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 17.1iT - 97T^{2} \) |
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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
−10.95050353983595811684543447311, −9.747651183367636924258770992413, −9.142617307290678735702265436298, −8.459963673589397238610113998112, −7.35510811125121330239917617707, −6.64038264448595145993936323192, −5.79125992895673267580862034034, −4.07391629183749039378944185865, −2.96694339552429525386792890186, −1.48624729291320279659259017351,
2.06052713283594362222649557710, 3.10753057982716449858673417019, 3.69978391233959430780309041522, 5.32866837452909263511883599662, 6.29224526715608071177875932201, 7.928545879975992779003835150733, 8.927161063688924610027388157290, 9.332224150654804292863371447800, 10.24989678632307386912326316119, 10.67336626772290880312089625464