L(s) = 1 | + (0.573 + 0.819i)2-s + (0.133 + 1.52i)3-s + (−0.342 + 0.939i)4-s + (−1.17 + 0.984i)6-s + (−0.227 + 0.848i)7-s + (−0.965 + 0.258i)8-s + (0.642 − 0.113i)9-s + (−1.39 + 2.40i)11-s + (−1.47 − 0.396i)12-s + (−5.85 − 0.512i)13-s + (−0.825 + 0.300i)14-s + (−0.766 − 0.642i)16-s + (−0.437 + 0.306i)17-s + (0.461 + 0.461i)18-s + (3.66 + 2.35i)19-s + ⋯ |
L(s) = 1 | + (0.405 + 0.579i)2-s + (0.0770 + 0.881i)3-s + (−0.171 + 0.469i)4-s + (−0.479 + 0.402i)6-s + (−0.0859 + 0.320i)7-s + (−0.341 + 0.0915i)8-s + (0.214 − 0.0377i)9-s + (−0.419 + 0.726i)11-s + (−0.427 − 0.114i)12-s + (−1.62 − 0.142i)13-s + (−0.220 + 0.0802i)14-s + (−0.191 − 0.160i)16-s + (−0.106 + 0.0742i)17-s + (0.108 + 0.108i)18-s + (0.841 + 0.539i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.102084 - 1.37683i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.102084 - 1.37683i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.573 - 0.819i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.66 - 2.35i)T \) |
good | 3 | \( 1 + (-0.133 - 1.52i)T + (-2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (0.227 - 0.848i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.39 - 2.40i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (5.85 + 0.512i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (0.437 - 0.306i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (5.86 - 2.73i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (0.259 + 1.47i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.95 - 2.28i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.22 + 4.22i)T - 37iT^{2} \) |
| 41 | \( 1 + (-2.19 + 2.61i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.10 - 6.64i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-1.04 + 1.49i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.40 - 3.01i)T + (-34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-1.12 + 6.39i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (0.643 + 0.234i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.786 - 0.550i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (1.03 + 2.83i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (9.60 - 0.840i)T + (71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (1.70 + 1.42i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-12.7 - 3.41i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (10.4 - 8.78i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.87 - 5.53i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−10.02670430141437717854974466409, −9.833859035859042645147794981065, −8.933797460587991137555308364959, −7.62468943945739209574325482438, −7.34818682273832514486573487848, −5.98308872161829804967662086108, −5.13320410280293329194542719818, −4.45806228583820642367575834285, −3.48028390932417994252907491051, −2.23023975760734653740955403073,
0.53165818888211182601071581401, 2.00998133390134536115662557048, 2.89016357768162554588386058809, 4.19363406592757181735569401955, 5.12925208083235735532278253675, 6.14714297832563281015995944467, 7.17566195112221025432058037286, 7.68900701850383559087703487077, 8.800164278046086180562736801877, 9.879764265974581769598352601265