| L(s) = 1 | + (2.04 + 1.39i)3-s + (−2.31 − 0.173i)5-s + (−2.63 − 0.205i)7-s + (1.13 + 2.89i)9-s + (−3.40 − 1.33i)11-s + (−3.01 + 2.40i)13-s + (−4.48 − 3.57i)15-s + (2.47 + 2.66i)17-s + (−0.709 + 1.22i)19-s + (−5.10 − 4.09i)21-s + (−5.93 + 6.40i)23-s + (0.370 + 0.0557i)25-s + (−0.0613 + 0.268i)27-s + (−0.774 − 3.39i)29-s + (2.01 + 3.48i)31-s + ⋯ |
| L(s) = 1 | + (1.17 + 0.804i)3-s + (−1.03 − 0.0774i)5-s + (−0.996 − 0.0777i)7-s + (0.378 + 0.965i)9-s + (−1.02 − 0.402i)11-s + (−0.834 + 0.665i)13-s + (−1.15 − 0.922i)15-s + (0.600 + 0.647i)17-s + (−0.162 + 0.282i)19-s + (−1.11 − 0.893i)21-s + (−1.23 + 1.33i)23-s + (0.0740 + 0.0111i)25-s + (−0.0118 + 0.0517i)27-s + (−0.143 − 0.629i)29-s + (0.361 + 0.625i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0308756 + 0.604081i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0308756 + 0.604081i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.63 + 0.205i)T \) |
| good | 3 | \( 1 + (-2.04 - 1.39i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (2.31 + 0.173i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (3.40 + 1.33i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (3.01 - 2.40i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.47 - 2.66i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.709 - 1.22i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.93 - 6.40i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.774 + 3.39i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.01 - 3.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.259 + 0.0800i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (0.974 + 2.02i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (3.12 - 6.47i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (2.87 - 0.433i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-9.23 + 2.84i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.0799 + 1.06i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-4.08 + 13.2i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-7.12 + 4.11i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (10.9 + 2.51i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.904 - 6.00i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-11.7 - 6.79i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.68 + 2.11i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.45 + 1.74i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 8.51iT - 97T^{2} \) |
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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
−10.28943908101584079062098014408, −9.883356105634918808789188796900, −9.067891738337380978886505652962, −7.984918819885783990475497403307, −7.78181099098674501211473836580, −6.41564249193900370194033946412, −5.14465560074386611453399434589, −3.89151127592272916487009740725, −3.53483619421747851503552343692, −2.39233256665866974082300271299,
0.24075403639356296611291882893, 2.41301689920024136229076633934, 2.99962476455642542840581810996, 4.10298951695575510747146782727, 5.42769371567391717260963019485, 6.79929087555609163572944119723, 7.48434607775025967786093382384, 8.009478701707309916740100775493, 8.796079804830224044100236226297, 9.884625031365297519426986689280
