L(s) = 1 | + (1.88 + 1.50i)2-s + (2.35 + 0.537i)3-s + (0.851 + 3.73i)4-s + (0.623 − 0.781i)5-s + (3.63 + 4.55i)6-s + (0.629 − 2.75i)7-s + (−1.91 + 3.97i)8-s + (2.54 + 1.22i)9-s + (2.35 − 0.537i)10-s + (−0.179 − 0.373i)11-s + 9.24i·12-s + (−3.44 + 1.66i)13-s + (5.33 − 4.25i)14-s + (1.88 − 1.50i)15-s + (−2.70 + 1.30i)16-s − 0.828i·17-s + ⋯ |
L(s) = 1 | + (1.33 + 1.06i)2-s + (1.35 + 0.310i)3-s + (0.425 + 1.86i)4-s + (0.278 − 0.349i)5-s + (1.48 + 1.86i)6-s + (0.237 − 1.04i)7-s + (−0.677 + 1.40i)8-s + (0.849 + 0.409i)9-s + (0.744 − 0.169i)10-s + (−0.0541 − 0.112i)11-s + 2.66i·12-s + (−0.956 + 0.460i)13-s + (1.42 − 1.13i)14-s + (0.487 − 0.388i)15-s + (−0.675 + 0.325i)16-s − 0.200i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.61741 + 3.20576i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.61741 + 3.20576i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (-1.88 - 1.50i)T + (0.445 + 1.94i)T^{2} \) |
| 3 | \( 1 + (-2.35 - 0.537i)T + (2.70 + 1.30i)T^{2} \) |
| 5 | \( 1 + (-0.623 + 0.781i)T + (-1.11 - 4.87i)T^{2} \) |
| 7 | \( 1 + (-0.629 + 2.75i)T + (-6.30 - 3.03i)T^{2} \) |
| 11 | \( 1 + (0.179 + 0.373i)T + (-6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (3.44 - 1.66i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 + (5.84 - 1.33i)T + (17.1 - 8.24i)T^{2} \) |
| 23 | \( 1 + (-2.28 - 2.85i)T + (-5.11 + 22.4i)T^{2} \) |
| 31 | \( 1 + (7.87 + 6.27i)T + (6.89 + 30.2i)T^{2} \) |
| 37 | \( 1 + (1.73 - 3.60i)T + (-23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + 4.48iT - 41T^{2} \) |
| 43 | \( 1 + (-2.80 + 2.23i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-1.40 - 2.92i)T + (-29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-5.91 + 7.41i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + (4.70 + 1.07i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + (-5.09 - 2.45i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (7.95 - 3.83i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (3.12 - 2.49i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + (-1.04 + 2.17i)T + (-49.2 - 61.7i)T^{2} \) |
| 83 | \( 1 + (1.70 + 7.46i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-9.76 - 7.78i)T + (19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + (-4.37 + 0.998i)T + (87.3 - 42.0i)T^{2} \) |
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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
−10.22043641843770688698015646701, −9.284347553272043987608532159650, −8.508866761820175972085502010084, −7.46868522337646355860608655616, −7.21523278002160156568242134181, −5.92280277588152029124310725184, −4.87847712029804537895414477540, −4.11918533764253722950617541115, −3.43571702097507796280295127588, −2.14269768273526532893404942033,
1.95149536725459685652071588749, 2.49051656211348586196167346348, 3.17611429133733178584901011323, 4.37074839047684555623437926538, 5.29595134064849361606805988941, 6.27996222083041315324102857702, 7.42414502166723461733053853060, 8.553461619429154219068455630045, 9.148734981480954139235565909001, 10.28704969109148338200249163553