L(s) = 1 | + (−0.125 − 1.99i)2-s + (−3.96 + 0.5i)4-s + (4.94 + 4.94i)7-s + (1.49 + 7.85i)8-s + (−6.36 + 6.36i)9-s + (−5.22 + 12.6i)11-s + (9.26 − 10.5i)14-s + (15.5 − 3.96i)16-s + (13.5 + 11.9i)18-s + (25.8 + 8.84i)22-s + (12.1 − 12.1i)23-s + (17.6 + 17.6i)25-s + (−22.1 − 17.1i)28-s + (22.1 + 53.5i)29-s + (−9.86 − 30.4i)32-s + ⋯ |
L(s) = 1 | + (−0.0626 − 0.998i)2-s + (−0.992 + 0.125i)4-s + (0.707 + 0.707i)7-s + (0.186 + 0.982i)8-s + (−0.707 + 0.707i)9-s + (−0.474 + 1.14i)11-s + (0.661 − 0.750i)14-s + (0.968 − 0.248i)16-s + (0.750 + 0.661i)18-s + (1.17 + 0.401i)22-s + (0.528 − 0.528i)23-s + (0.707 + 0.707i)25-s + (−0.789 − 0.613i)28-s + (0.765 + 1.84i)29-s + (−0.308 − 0.951i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07564 + 0.280202i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07564 + 0.280202i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.125 + 1.99i)T \) |
| 7 | \( 1 + (-4.94 - 4.94i)T \) |
good | 3 | \( 1 + (6.36 - 6.36i)T^{2} \) |
| 5 | \( 1 + (-17.6 - 17.6i)T^{2} \) |
| 11 | \( 1 + (5.22 - 12.6i)T + (-85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (-119. + 119. i)T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-12.1 + 12.1i)T - 529iT^{2} \) |
| 29 | \( 1 + (-22.1 - 53.5i)T + (-594. + 594. i)T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + (67.6 + 28.0i)T + (968. + 968. i)T^{2} \) |
| 41 | \( 1 + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-13.9 + 33.6i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + (36.5 - 88.2i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (39.7 + 95.9i)T + (-3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (-59.8 - 59.8i)T + 5.04e3iT^{2} \) |
| 73 | \( 1 + 5.32e3iT^{2} \) |
| 79 | \( 1 - 23.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 - 7.92e3iT^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−12.20543728392601134962346700516, −10.97924902398272152857633817516, −10.52995605045434347675397552454, −9.130068978928691004382501489876, −8.486592388022060588268116775284, −7.31836950272198574316407771636, −5.35459421681946259652703290606, −4.76365539647367091052042110830, −2.95829285055756063958438994847, −1.81881636968344228230828217856,
0.64497943976343179949334934172, 3.37365948929583932237905658919, 4.73764933412013642423289347243, 5.84808145296909447629546563805, 6.82846255341272509802982864586, 8.090155390811512978557883958173, 8.599155014470524383643794810041, 9.861672426173572942774274440298, 10.91900711677508792985880485279, 11.93666597528484827744846432791