| L(s) = 1 | + (0.888 + 0.458i)2-s + (−1.97 + 2.51i)3-s + (0.580 + 0.814i)4-s + (0.841 + 0.802i)5-s + (−2.90 + 1.32i)6-s + (−1.02 + 2.43i)7-s + (0.142 + 0.989i)8-s + (−1.70 − 7.00i)9-s + (0.380 + 1.09i)10-s + (−0.334 − 0.648i)11-s + (−3.19 − 0.152i)12-s + (0.469 + 0.406i)13-s + (−2.03 + 1.69i)14-s + (−3.67 + 0.528i)15-s + (−0.327 + 0.945i)16-s + (3.54 + 0.338i)17-s + ⋯ |
| L(s) = 1 | + (0.628 + 0.324i)2-s + (−1.14 + 1.45i)3-s + (0.290 + 0.407i)4-s + (0.376 + 0.358i)5-s + (−1.18 + 0.541i)6-s + (−0.388 + 0.921i)7-s + (0.0503 + 0.349i)8-s + (−0.566 − 2.33i)9-s + (0.120 + 0.347i)10-s + (−0.100 − 0.195i)11-s + (−0.921 − 0.0438i)12-s + (0.130 + 0.112i)13-s + (−0.542 + 0.453i)14-s + (−0.949 + 0.136i)15-s + (−0.0817 + 0.236i)16-s + (0.860 + 0.0821i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.121414 + 1.16530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121414 + 1.16530i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.888 - 0.458i)T \) |
| 7 | \( 1 + (1.02 - 2.43i)T \) |
| 23 | \( 1 + (3.90 + 2.78i)T \) |
| good | 3 | \( 1 + (1.97 - 2.51i)T + (-0.707 - 2.91i)T^{2} \) |
| 5 | \( 1 + (-0.841 - 0.802i)T + (0.237 + 4.99i)T^{2} \) |
| 11 | \( 1 + (0.334 + 0.648i)T + (-6.38 + 8.96i)T^{2} \) |
| 13 | \( 1 + (-0.469 - 0.406i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.54 - 0.338i)T + (16.6 + 3.21i)T^{2} \) |
| 19 | \( 1 + (2.38 - 0.228i)T + (18.6 - 3.59i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 5.39i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (3.52 - 8.80i)T + (-22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (-0.488 + 0.118i)T + (32.8 - 16.9i)T^{2} \) |
| 41 | \( 1 + (-2.57 - 8.78i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-10.6 - 1.53i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-6.07 + 3.50i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.366 - 1.90i)T + (-49.2 + 19.6i)T^{2} \) |
| 59 | \( 1 + (-1.18 + 0.411i)T + (46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (0.688 - 0.541i)T + (14.3 - 59.2i)T^{2} \) |
| 67 | \( 1 + (-7.39 + 0.352i)T + (66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (5.42 - 3.48i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-8.50 + 6.05i)T + (23.8 - 68.9i)T^{2} \) |
| 79 | \( 1 + (-0.270 + 1.40i)T + (-73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (-8.40 - 2.46i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (13.5 - 5.44i)T + (64.4 - 61.4i)T^{2} \) |
| 97 | \( 1 + (17.0 - 5.01i)T + (81.6 - 52.4i)T^{2} \) |
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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
−12.18004393887914199452478930219, −11.02104845312170966936881506581, −10.39239915790592664509868668852, −9.485862435659857884214264699825, −8.475627257403371558175961689023, −6.64146045115319664583480433049, −5.93406393293296178958079799098, −5.23779113571795292271766308146, −4.13348083427251521666044080042, −2.95051863717793561631293279163,
0.798855661343717533568383146363, 2.15475451149406755553592484313, 4.09648737143438519848790803703, 5.52961156237274161880441996936, 6.05583271770946962655409605002, 7.18747511068769774045590908134, 7.82836193963834361711660880322, 9.628800919410920539372623996153, 10.64662575926050329928779593406, 11.38471776689536565600312978487
