| L(s) = 1 | + (1.39 + 0.228i)2-s + (1.12 + 0.467i)3-s + (1.89 + 0.638i)4-s + (−0.906 + 0.462i)5-s + (1.46 + 0.910i)6-s + (−1.47 − 2.41i)7-s + (2.49 + 1.32i)8-s + (−1.06 − 1.06i)9-s + (−1.37 + 0.437i)10-s + (−5.44 − 0.428i)11-s + (1.84 + 1.60i)12-s + (1.37 + 5.73i)13-s + (−1.50 − 3.70i)14-s + (−1.23 + 0.0975i)15-s + (3.18 + 2.42i)16-s + (5.06 − 4.32i)17-s + ⋯ |
| L(s) = 1 | + (0.986 + 0.161i)2-s + (0.651 + 0.269i)3-s + (0.947 + 0.319i)4-s + (−0.405 + 0.206i)5-s + (0.599 + 0.371i)6-s + (−0.558 − 0.910i)7-s + (0.883 + 0.468i)8-s + (−0.355 − 0.355i)9-s + (−0.433 + 0.138i)10-s + (−1.64 − 0.129i)11-s + (0.531 + 0.463i)12-s + (0.381 + 1.59i)13-s + (−0.403 − 0.989i)14-s + (−0.320 + 0.0251i)15-s + (0.796 + 0.605i)16-s + (1.22 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.00425 + 0.392308i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00425 + 0.392308i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 - 0.228i)T \) |
| 41 | \( 1 + (6.40 - 0.00937i)T \) |
| good | 3 | \( 1 + (-1.12 - 0.467i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (0.906 - 0.462i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (1.47 + 2.41i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (5.44 + 0.428i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (-1.37 - 5.73i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-5.06 + 4.32i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (3.80 + 0.912i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (-4.52 + 3.29i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.13 - 0.968i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-2.34 - 7.21i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.577 + 1.77i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (0.547 + 0.0867i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.86 + 3.04i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (4.02 - 4.71i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (0.796 + 1.09i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.44 + 1.33i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.660 - 8.39i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-0.307 + 3.91i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-2.33 + 2.33i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.36 - 5.70i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 1.36iT - 83T^{2} \) |
| 89 | \( 1 + (-12.4 + 7.61i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (0.242 + 3.08i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−13.18834524839518578444178234947, −12.04745109302207537123094536751, −11.03139036530421566872434345203, −10.07735478240059041716469148931, −8.647374429192137375148844482423, −7.45764849314847964265001800671, −6.60532868462875187525713379116, −5.03604115693895658096784339415, −3.75804429068013477185158258245, −2.82276363718890077940949342179,
2.48552594060860235070752524042, 3.38329747113635031497631925282, 5.24659112925083930470831634795, 5.96878813306103822347015494182, 7.84831889228328121695364389543, 8.192017165414336556477139800077, 10.03558017093821761611382193198, 10.86831280194093233390357652538, 12.19928612279741370212122808494, 12.97442372216343335093834887924
