| L(s) = 1 | + (−0.0525 + 0.247i)2-s + (0.690 + 0.766i)3-s + (1.76 + 0.787i)4-s + 3.00i·5-s + (−0.225 + 0.130i)6-s + (−0.278 + 0.624i)7-s + (−0.585 + 0.805i)8-s + (0.202 − 1.92i)9-s + (−0.744 − 0.158i)10-s + (−0.385 + 0.0405i)11-s + (0.617 + 1.89i)12-s + (−0.696 + 3.53i)13-s + (−0.139 − 0.101i)14-s + (−2.30 + 2.07i)15-s + (2.42 + 2.69i)16-s + (0.685 − 6.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.0371 + 0.174i)2-s + (0.398 + 0.442i)3-s + (0.884 + 0.393i)4-s + 1.34i·5-s + (−0.0922 + 0.0532i)6-s + (−0.105 + 0.236i)7-s + (−0.206 + 0.284i)8-s + (0.0674 − 0.641i)9-s + (−0.235 − 0.0500i)10-s + (−0.116 + 0.0122i)11-s + (0.178 + 0.548i)12-s + (−0.193 + 0.981i)13-s + (−0.0373 − 0.0271i)14-s + (−0.595 + 0.536i)15-s + (0.605 + 0.672i)16-s + (0.166 − 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0822 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0822 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20226 + 1.30555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20226 + 1.30555i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (0.696 - 3.53i)T \) |
| 31 | \( 1 + (-4.78 - 2.84i)T \) |
| good | 2 | \( 1 + (0.0525 - 0.247i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.690 - 0.766i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 - 3.00iT - 5T^{2} \) |
| 7 | \( 1 + (0.278 - 0.624i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.385 - 0.0405i)T + (10.7 - 2.28i)T^{2} \) |
| 17 | \( 1 + (-0.685 + 6.51i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (4.45 + 4.01i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.54 + 1.13i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (7.49 + 1.59i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + (-9.45 - 5.45i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.546 - 2.57i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.40 + 2.67i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-6.78 - 2.20i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (6.94 + 5.04i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.854 + 4.02i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-4.29 - 7.44i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.54 + 1.47i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (12.7 + 1.34i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (6.80 + 9.36i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-6.68 - 4.85i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-12.5 + 4.07i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.304 + 0.0320i)T + (87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (-7.05 + 15.8i)T + (-64.9 - 72.0i)T^{2} \) |
| 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
−11.42050773524384490321056864822, −10.72080642310428651459697947382, −9.642129304712896581749392580588, −8.880095071403309843335789807039, −7.52168696221751379413065378731, −6.83665364710280755867291630980, −6.19172750046727322628360413069, −4.43596277529892481244366086055, −3.08202163001819390233842218123, −2.51482632154217371632235130827,
1.23200360845820885315107633822, 2.33685121537107996248783403237, 3.96228575449316754364207780418, 5.35499304748445624600140493258, 6.13921740377448890294410717823, 7.59759326141283034189903292614, 8.079536968592219486069148986091, 9.129191915846383385502448061561, 10.33536699281811759596270756496, 10.82658608273829681946256662452
