| L(s) = 1 | + (0.532 + 2.07i)2-s + (−2.80 + 0.535i)3-s + (−2.26 + 1.24i)4-s + (−2.60 − 5.53i)6-s + (1.69 − 2.33i)7-s + (−0.862 − 0.918i)8-s + (4.80 − 1.90i)9-s + (5.22 − 1.34i)11-s + (5.69 − 4.71i)12-s + (−0.412 − 1.04i)13-s + (5.75 + 2.27i)14-s + (−1.32 + 2.09i)16-s + (2.22 − 4.05i)17-s + (6.50 + 8.95i)18-s + (0.221 − 1.16i)19-s + ⋯ |
| L(s) = 1 | + (0.376 + 1.46i)2-s + (−1.62 + 0.309i)3-s + (−1.13 + 0.623i)4-s + (−1.06 − 2.26i)6-s + (0.641 − 0.883i)7-s + (−0.304 − 0.324i)8-s + (1.60 − 0.633i)9-s + (1.57 − 0.404i)11-s + (1.64 − 1.36i)12-s + (−0.114 − 0.288i)13-s + (1.53 + 0.608i)14-s + (−0.331 + 0.522i)16-s + (0.540 − 0.983i)17-s + (1.53 + 2.10i)18-s + (0.0508 − 0.266i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0218 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0218 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.863059 + 0.844362i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.863059 + 0.844362i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-0.532 - 2.07i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (2.80 - 0.535i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (-1.69 + 2.33i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-5.22 + 1.34i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (0.412 + 1.04i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-2.22 + 4.05i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.221 + 1.16i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (2.10 + 0.132i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (5.27 + 0.666i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (-4.18 - 2.29i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (-6.38 - 4.05i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.295 + 4.69i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-10.1 + 3.28i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (1.68 - 1.79i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-2.60 - 1.22i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (-0.690 - 0.835i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (-0.178 + 2.84i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (-1.77 - 14.0i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (1.69 + 1.59i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (9.66 + 7.99i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (-1.42 - 7.44i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (-14.2 - 2.70i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-0.396 + 0.479i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-1.18 + 9.39i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−11.02853228501600266525953588456, −10.01284868925803298726001593470, −8.961624983592373889689894590917, −7.68798988357211142564776982694, −7.03491159207942818419320177127, −6.23195972304145326774135451046, −5.54303186206066425162372425254, −4.63844901579236119510228898859, −3.98830692753892178652470231970, −0.958746659281878776086821040501,
1.18254545487248580624816748967, 2.05907327261912134253963780536, 3.87592430304132356405541184670, 4.65047823887055741584242218019, 5.71225097590414202373048583825, 6.41547868250797455582268422457, 7.65262742919443278726596192861, 9.101640388035932228337754959013, 9.889281833136837017506883937908, 10.81594353906112511206373557003
