| L(s) = 1 | + (0.168 − 0.156i)2-s + (0.223 − 0.0336i)3-s + (−0.145 + 1.94i)4-s + (0.711 − 1.81i)5-s + (0.0322 − 0.0404i)6-s + (−2.04 − 1.68i)7-s + (0.564 + 0.708i)8-s + (−2.81 + 0.869i)9-s + (−0.163 − 0.415i)10-s + (−1.25 − 0.387i)11-s + (0.0328 + 0.438i)12-s + (0.866 + 3.79i)13-s + (−0.606 + 0.0357i)14-s + (0.0977 − 0.428i)15-s + (−3.64 − 0.549i)16-s + (5.18 − 3.53i)17-s + ⋯ |
| L(s) = 1 | + (0.118 − 0.110i)2-s + (0.128 − 0.0194i)3-s + (−0.0727 + 0.970i)4-s + (0.318 − 0.810i)5-s + (0.0131 − 0.0165i)6-s + (−0.771 − 0.635i)7-s + (0.199 + 0.250i)8-s + (−0.939 + 0.289i)9-s + (−0.0516 − 0.131i)10-s + (−0.379 − 0.116i)11-s + (0.00948 + 0.126i)12-s + (0.240 + 1.05i)13-s + (−0.162 + 0.00954i)14-s + (0.0252 − 0.110i)15-s + (−0.911 − 0.137i)16-s + (1.25 − 0.857i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.853080 + 0.00462739i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.853080 + 0.00462739i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.04 + 1.68i)T \) |
| good | 2 | \( 1 + (-0.168 + 0.156i)T + (0.149 - 1.99i)T^{2} \) |
| 3 | \( 1 + (-0.223 + 0.0336i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (-0.711 + 1.81i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (1.25 + 0.387i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 3.79i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-5.18 + 3.53i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (1.89 + 3.28i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.21 - 2.19i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-5.98 - 2.88i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (0.842 - 1.45i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0956 + 1.27i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (5.27 + 6.61i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (3.86 - 4.84i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (2.40 - 2.23i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.377 + 5.03i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (4.20 + 10.7i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.0815 - 1.08i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-4.35 + 7.53i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (13.6 - 6.58i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-9.63 - 8.94i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-2.17 - 3.77i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.19 + 9.63i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (4.32 - 1.33i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−16.10943933475379394767963929234, −14.08928949996623511330592402336, −13.36825960358568845783685574944, −12.36232999985189270929802512107, −11.16528665776697209445941907651, −9.431360420118711124930793311064, −8.429585844492767424195895190287, −6.97818617772848618470080104928, −4.99108888216298281730296985396, −3.17936965716638968744357053786,
2.97901751128622518920813323525, 5.61672194355039150115459987908, 6.39207695123462652967145133288, 8.404676431520458284550782221223, 9.945683304924477496637089963615, 10.61535638682896866514436838845, 12.26374135537257763215976920566, 13.58204447839463558103628816544, 14.79898047944237073633154286078, 15.15909544462321907698238443668
