| L(s) = 1 | + (−0.951 − 0.309i)2-s + (1.95 − 2.69i)3-s + (0.809 + 0.587i)4-s + (2.12 + 0.708i)5-s + (−2.69 + 1.95i)6-s − i·7-s + (−0.587 − 0.809i)8-s + (−2.50 − 7.71i)9-s + (−1.79 − 1.32i)10-s + (−0.532 + 1.63i)11-s + (3.17 − 1.03i)12-s + (−0.812 + 0.263i)13-s + (−0.309 + 0.951i)14-s + (6.06 − 4.33i)15-s + (0.309 + 0.951i)16-s + (1.75 + 2.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 − 0.218i)2-s + (1.13 − 1.55i)3-s + (0.404 + 0.293i)4-s + (0.948 + 0.316i)5-s + (−1.10 + 0.800i)6-s − 0.377i·7-s + (−0.207 − 0.286i)8-s + (−0.836 − 2.57i)9-s + (−0.568 − 0.420i)10-s + (−0.160 + 0.493i)11-s + (0.915 − 0.297i)12-s + (−0.225 + 0.0732i)13-s + (−0.0825 + 0.254i)14-s + (1.56 − 1.11i)15-s + (0.0772 + 0.237i)16-s + (0.425 + 0.585i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0795 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0795 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06369 - 1.15196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06369 - 1.15196i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 + (-2.12 - 0.708i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 + (-1.95 + 2.69i)T + (-0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (0.532 - 1.63i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.812 - 0.263i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.75 - 2.41i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.65 - 1.20i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-7.59 - 2.46i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (7.36 + 5.34i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (3.44 - 2.50i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (4.49 - 1.46i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.92 + 5.91i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 12.6iT - 43T^{2} \) |
| 47 | \( 1 + (-3.87 + 5.33i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.30 + 4.55i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.93 - 12.1i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.847 + 2.60i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.84 - 2.54i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-5.91 - 4.30i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.91 + 2.89i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.31 + 0.954i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.850 - 1.17i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.620 - 1.91i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (3.60 - 4.95i)T + (-29.9 - 92.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.24528508714781131918398795854, −10.05857367952079671558607287664, −9.255174724156457941338674476904, −8.451054349987596519123781546250, −7.36911995057387717708768790077, −6.95372764856689166956607849893, −5.79611160330219889450229226115, −3.46256086156995659553120500683, −2.31795523833940309856729154459, −1.40858855726451103869602445657,
2.25085158501838784588974691235, 3.31092242866662108308724339496, 4.92040883054598793214709573717, 5.58369452098269616342849075339, 7.26108772858204405943337737027, 8.552555572941948412513241715891, 9.017597960552025899053775100805, 9.616637456404380337165767696765, 10.51781413187430876388170183840, 11.13153602603823077359018062443
