L(s) = 1 | + (1.37 − 0.309i)2-s + (0.821 + 0.220i)3-s + (1.80 − 0.854i)4-s + (−0.801 − 2.08i)5-s + (1.20 + 0.0493i)6-s + (1.14 + 2.38i)7-s + (2.23 − 1.73i)8-s + (−1.97 − 1.13i)9-s + (−1.75 − 2.63i)10-s + (0.933 + 1.61i)11-s + (1.67 − 0.303i)12-s + (−1.15 − 1.15i)13-s + (2.31 + 2.93i)14-s + (−0.198 − 1.89i)15-s + (2.53 − 3.09i)16-s + (−1.40 + 5.24i)17-s + ⋯ |
L(s) = 1 | + (0.975 − 0.218i)2-s + (0.474 + 0.127i)3-s + (0.904 − 0.427i)4-s + (−0.358 − 0.933i)5-s + (0.490 + 0.0201i)6-s + (0.432 + 0.901i)7-s + (0.788 − 0.614i)8-s + (−0.657 − 0.379i)9-s + (−0.553 − 0.832i)10-s + (0.281 + 0.487i)11-s + (0.482 − 0.0877i)12-s + (−0.321 − 0.321i)13-s + (0.619 + 0.784i)14-s + (−0.0512 − 0.488i)15-s + (0.634 − 0.772i)16-s + (−0.340 + 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.34477 - 0.630090i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.34477 - 0.630090i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 + 0.309i)T \) |
| 5 | \( 1 + (0.801 + 2.08i)T \) |
| 7 | \( 1 + (-1.14 - 2.38i)T \) |
good | 3 | \( 1 + (-0.821 - 0.220i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-0.933 - 1.61i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.15 + 1.15i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.40 - 5.24i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.58 + 0.915i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.26 + 0.605i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 + (5.06 - 2.92i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.55 - 9.54i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 4.49T + 41T^{2} \) |
| 43 | \( 1 + (1.18 - 1.18i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.20 + 4.50i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.06 + 11.4i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-7.77 + 4.48i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.16 + 4.71i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.871 + 3.25i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 3.09iT - 71T^{2} \) |
| 73 | \( 1 + (8.12 + 2.17i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.84 - 10.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.95 + 4.95i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.31 - 4.22i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.17 - 6.17i)T + 97iT^{2} \) |
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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
−11.98612959942090625850268980351, −11.22391027616429258349410285767, −9.913937519191640477642527902578, −8.770537208875468867820673616654, −8.117258919263289365632649899305, −6.58304954905047105563690958780, −5.44667140920240991653695735390, −4.56547346062970652121995019972, −3.33546375965161289135942151697, −1.90263094838562537196934251458,
2.40378618075371598271773551167, 3.48284970333419029359788788795, 4.59413981831390907440932294353, 5.94995481315886163055896607818, 7.19808888841949925269049069480, 7.58999247701212373732030332673, 8.859676782846005419668799655850, 10.46388870344734018822636483859, 11.22281492264054836813037739286, 11.77560836578151588003242624097