| L(s) = 1 | + (0.973 − 0.230i)2-s + (0.190 − 1.72i)3-s + (0.893 − 0.448i)4-s + (−1.06 − 1.43i)5-s + (−0.211 − 1.71i)6-s + (−0.983 + 0.646i)7-s + (0.766 − 0.642i)8-s + (−2.92 − 0.655i)9-s + (−1.37 − 1.15i)10-s + (3.61 + 0.422i)11-s + (−0.602 − 1.62i)12-s + (1.84 + 6.16i)13-s + (−0.807 + 0.856i)14-s + (−2.67 + 1.56i)15-s + (0.597 − 0.802i)16-s + (0.461 − 2.61i)17-s + ⋯ |
| L(s) = 1 | + (0.688 − 0.163i)2-s + (0.109 − 0.993i)3-s + (0.446 − 0.224i)4-s + (−0.478 − 0.642i)5-s + (−0.0864 − 0.701i)6-s + (−0.371 + 0.244i)7-s + (0.270 − 0.227i)8-s + (−0.975 − 0.218i)9-s + (−0.434 − 0.364i)10-s + (1.09 + 0.127i)11-s + (−0.173 − 0.468i)12-s + (0.511 + 1.71i)13-s + (−0.215 + 0.228i)14-s + (−0.691 + 0.404i)15-s + (0.149 − 0.200i)16-s + (0.111 − 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25215 - 0.945737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25215 - 0.945737i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.973 + 0.230i)T \) |
| 3 | \( 1 + (-0.190 + 1.72i)T \) |
| good | 5 | \( 1 + (1.06 + 1.43i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (0.983 - 0.646i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (-3.61 - 0.422i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (-1.84 - 6.16i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (-0.461 + 2.61i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.128 + 0.727i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-3.51 - 2.31i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (1.71 + 1.81i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (0.374 - 6.43i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (7.73 - 2.81i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (9.76 + 2.31i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (-2.11 - 4.91i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (0.160 + 2.74i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (-4.71 + 8.17i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (9.18 - 1.07i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (10.4 + 5.23i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (-1.44 + 1.53i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-8.87 - 7.44i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-6.70 + 5.62i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-5.84 + 1.38i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (0.953 - 0.226i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (4.23 - 3.54i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-10.4 + 14.0i)T + (-27.8 - 92.9i)T^{2} \) |
| 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
−12.50559136321712488826267632937, −11.94286181176122934057566658790, −11.25842508939035580951645428682, −9.350388183413540115377403576656, −8.588123167311123207745568924549, −7.05122280933103450808630244294, −6.41388088368488489547947609932, −4.86644624631334747387652599963, −3.47075054200475260403968993160, −1.61157395946751076879537954751,
3.21577376078491172057183244198, 3.84957459170748771250641877037, 5.38161730016939457377772566854, 6.49369976232570392874142220180, 7.83065366141144025902851945912, 9.003160387166899275761569685497, 10.40855155001181751522802274627, 10.92625902217510445441374033089, 12.04894337408222777686583198734, 13.18866069042051065128967139591
