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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.304324 - 0.357858i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.304324 - 0.357858i\) |
\(L(\frac{3}{2})\) |
|
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
−10.02122607813636707723587569164, −9.524751764427014563012520666359, −8.452165645061926324956262966530, −7.32606933821160318898687347642, −6.95862251161276008711624717861, −5.48160017266301799531313502776, −4.61089321462873755532434496267, −3.13252215477956549131480877625, −1.54743694664400488968375974637, −0.75618198234381513461313468474,
0.74618650975108476512765213925, 2.58714473508557471157493514455, 4.79933653257148156423393557791, 5.55778780629583880727950728327, 6.12264874605430507137313092820, 6.93990233135114037157980655149, 7.86889065824014341526188592033, 8.711297121112129120084554214187, 9.742441517290921978675908188082, 10.09760343392094699979110760799