| L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (0.101 − 0.577i)5-s + (−0.173 − 0.984i)6-s + (−0.5 − 2.59i)7-s + (0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.448 + 0.376i)10-s + (−1.40 − 2.43i)11-s + (−0.500 + 0.866i)12-s + (−0.176 − 1.00i)13-s + (−1.28 + 2.31i)14-s + (0.448 − 0.376i)15-s + (−0.939 + 0.342i)16-s + (0.498 − 2.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.442 + 0.371i)3-s + (0.0868 + 0.492i)4-s + (0.0455 − 0.258i)5-s + (−0.0708 − 0.402i)6-s + (−0.188 − 0.981i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.141 + 0.119i)10-s + (−0.424 − 0.734i)11-s + (−0.144 + 0.249i)12-s + (−0.0490 − 0.278i)13-s + (−0.343 + 0.617i)14-s + (0.115 − 0.0972i)15-s + (−0.234 + 0.0855i)16-s + (0.120 − 0.686i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.436801 - 0.823024i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.436801 - 0.823024i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| 19 | \( 1 + (4.18 - 1.23i)T \) |
| good | 5 | \( 1 + (-0.101 + 0.577i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (1.40 + 2.43i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.176 + 1.00i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.498 + 2.82i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (5.82 + 2.12i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.17 + 0.790i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + 1.37T + 31T^{2} \) |
| 37 | \( 1 + (3.41 + 5.91i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.53 + 8.70i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.01 - 5.88i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.0272 - 0.154i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (1.29 + 7.35i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.64 + 9.30i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.69 - 1.34i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 9.31i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.573 - 0.481i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-9.22 - 7.74i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (6.75 - 2.45i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (8.23 + 14.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.82 - 7.40i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.18 + 0.429i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.01768283935793351819411465382, −9.212388483638141847128359918870, −8.351514531728947521284570993547, −7.69611020236791615141348175958, −6.69988399378381380498713602460, −5.42205527944959746253106364715, −4.20919899181862254078247297317, −3.42033439959907426854391342781, −2.19677464368780147038985556111, −0.50440721939578096738105237257,
1.81179482853211005535481976031, 2.72769574613863933144576845660, 4.25198790380893482773190000330, 5.52412858068869219394064374516, 6.37472485093848283281479767326, 7.15203728496836376947419560552, 8.137504710955922182205320743979, 8.710574536202410970802428911223, 9.600472887343996191518517298405, 10.30114901917934786438786290609
