| 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 | \( 1 + (-10.4 - 14.0i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−13.18866069042051065128967139591, −12.04894337408222777686583198734, −10.92625902217510445441374033089, −10.40855155001181751522802274627, −9.003160387166899275761569685497, −7.83065366141144025902851945912, −6.49369976232570392874142220180, −5.38161730016939457377772566854, −3.84957459170748771250641877037, −3.21577376078491172057183244198,
1.61157395946751076879537954751, 3.47075054200475260403968993160, 4.86644624631334747387652599963, 6.41388088368488489547947609932, 7.05122280933103450808630244294, 8.588123167311123207745568924549, 9.350388183413540115377403576656, 11.25842508939035580951645428682, 11.94286181176122934057566658790, 12.50559136321712488826267632937
