| L(s) = 1 | + (−1.35 − 0.405i)2-s + (0.810 + 1.53i)3-s + (1.67 + 1.09i)4-s + (−0.190 + 0.560i)5-s + (−0.476 − 2.40i)6-s + (−1.38 − 1.05i)7-s + (−1.81 − 2.16i)8-s + (−1.68 + 2.48i)9-s + (0.485 − 0.682i)10-s + (−0.168 + 2.57i)11-s + (−0.328 + 3.44i)12-s + (−4.98 − 4.36i)13-s + (1.44 + 1.99i)14-s + (−1.01 + 0.162i)15-s + (1.58 + 3.67i)16-s + (−3.24 + 3.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.958 − 0.286i)2-s + (0.467 + 0.883i)3-s + (0.835 + 0.549i)4-s + (−0.0851 + 0.250i)5-s + (−0.194 − 0.980i)6-s + (−0.522 − 0.400i)7-s + (−0.643 − 0.765i)8-s + (−0.562 + 0.826i)9-s + (0.153 − 0.215i)10-s + (−0.0508 + 0.775i)11-s + (−0.0947 + 0.995i)12-s + (−1.38 − 1.21i)13-s + (0.385 + 0.533i)14-s + (−0.261 + 0.0420i)15-s + (0.396 + 0.917i)16-s + (−0.788 + 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0318202 + 0.370768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0318202 + 0.370768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 + 0.405i)T \) |
| 3 | \( 1 + (-0.810 - 1.53i)T \) |
| good | 5 | \( 1 + (0.190 - 0.560i)T + (-3.96 - 3.04i)T^{2} \) |
| 7 | \( 1 + (1.38 + 1.05i)T + (1.81 + 6.76i)T^{2} \) |
| 11 | \( 1 + (0.168 - 2.57i)T + (-10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (4.98 + 4.36i)T + (1.69 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.24 - 3.24i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.0330 + 0.166i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (5.65 - 4.33i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (1.27 - 2.57i)T + (-17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (-3.97 + 6.88i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.20 - 0.240i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (3.83 + 4.99i)T + (-10.6 + 39.6i)T^{2} \) |
| 43 | \( 1 + (0.331 - 5.06i)T + (-42.6 - 5.61i)T^{2} \) |
| 47 | \( 1 + (-1.62 + 6.05i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.13 - 6.19i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-4.47 - 1.51i)T + (46.8 + 35.9i)T^{2} \) |
| 61 | \( 1 + (6.79 - 13.7i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (0.155 + 2.37i)T + (-66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (-3.86 - 9.32i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (4.40 - 10.6i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.73 + 6.46i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-0.817 - 2.40i)T + (-65.8 + 50.5i)T^{2} \) |
| 89 | \( 1 + (2.37 + 5.74i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (9.57 - 5.53i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.70892745513432450730501860796, −10.06711232053987468829674549469, −9.696617992691314888547779395904, −8.633862023521677575779640585315, −7.72402264010299628989301586042, −7.05943907032748481855894431184, −5.66600295631716992524956456797, −4.28240461397616285981008735940, −3.22615958350927506811244773609, −2.21758126072619112717303571364,
0.24447931125828047232123340439, 2.02552118008884100286770826870, 2.92729865776983629246357957608, 4.83815871715643720839752270717, 6.32269018353008673023884538177, 6.70149255637663016412620631200, 7.75017373186158207894534849539, 8.593920725233855743887465230845, 9.202178572123882179395253905785, 9.978659910863361460587100881019
