| L(s) = 1 | + (0.402 − 0.258i)2-s + (−0.313 + 2.18i)3-s + (−0.735 + 1.61i)4-s + (2.48 − 0.730i)5-s + (0.438 + 0.959i)6-s + (2.22 + 2.56i)7-s + (0.256 + 1.78i)8-s + (−1.78 − 0.522i)9-s + (0.813 − 0.938i)10-s + (0.192 + 0.123i)11-s + (−3.28 − 2.10i)12-s + (2.00 − 2.31i)13-s + (1.56 + 0.458i)14-s + (0.813 + 5.65i)15-s + (−1.75 − 2.02i)16-s + (−0.800 − 1.75i)17-s + ⋯ |
| L(s) = 1 | + (0.284 − 0.183i)2-s + (−0.181 + 1.25i)3-s + (−0.367 + 0.805i)4-s + (1.11 − 0.326i)5-s + (0.178 + 0.391i)6-s + (0.841 + 0.970i)7-s + (0.0908 + 0.631i)8-s + (−0.593 − 0.174i)9-s + (0.257 − 0.296i)10-s + (0.0580 + 0.0372i)11-s + (−0.947 − 0.608i)12-s + (0.555 − 0.641i)13-s + (0.417 + 0.122i)14-s + (0.210 + 1.46i)15-s + (−0.438 − 0.505i)16-s + (−0.194 − 0.425i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23152 + 1.43757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23152 + 1.43757i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (-0.402 + 0.258i)T + (0.830 - 1.81i)T^{2} \) |
| 3 | \( 1 + (0.313 - 2.18i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (-2.48 + 0.730i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-2.22 - 2.56i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.192 - 0.123i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.00 + 2.31i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.800 + 1.75i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.803 + 1.75i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.68 + 5.88i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.214 + 1.49i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-6.47 - 1.90i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (7.73 - 2.27i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.378 - 2.63i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 + (5.61 + 6.48i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.0404 - 0.0466i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.00632 - 0.0439i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-10.4 + 6.70i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (7.01 - 4.50i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.277 - 0.607i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-7.83 + 9.04i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.345 - 0.101i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.29 + 15.9i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.382 + 0.112i)T + (81.6 - 52.4i)T^{2} \) |
| 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
−11.26746305989105595861338754567, −10.05363958500511220919596830950, −9.367209681767449695784560027417, −8.690980178143584952334832028378, −7.80476438662293253279965498847, −6.05496167839485375008171679233, −5.16178506177891892798320903757, −4.65767122859948678280105197948, −3.38849450459171119061406582671, −2.14331877207019898111740573408,
1.23078077759071882129107289710, 1.89244317490055674598993903682, 3.99797522879227110562309303773, 5.20826273740487108515496107862, 6.16879809443444917197939987590, 6.73156832172785318483578762335, 7.63249668475250454337202810213, 8.804812818322438611333638886374, 9.856507901791982251570806959779, 10.63396157755813717861992331632
