| L(s) = 1 | + (0.654 − 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.454 − 0.995i)5-s + (0.142 − 0.989i)6-s + (0.959 − 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (−1.05 − 0.308i)10-s + (1.14 + 1.31i)11-s + (−0.654 − 0.755i)12-s + (−2.07 − 0.608i)13-s + (0.415 − 0.909i)14-s + (−0.920 − 0.591i)15-s + (−0.959 + 0.281i)16-s + (0.510 − 3.55i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.203 − 0.445i)5-s + (0.0580 − 0.404i)6-s + (0.362 − 0.106i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.332 − 0.0974i)10-s + (0.344 + 0.397i)11-s + (−0.189 − 0.218i)12-s + (−0.574 − 0.168i)13-s + (0.111 − 0.243i)14-s + (−0.237 − 0.152i)15-s + (−0.239 + 0.0704i)16-s + (0.123 − 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04557 - 1.95088i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04557 - 1.95088i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-2.00 + 4.35i)T \) |
| good | 5 | \( 1 + (0.454 + 0.995i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (-1.14 - 1.31i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.07 + 0.608i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.510 + 3.55i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.582 + 4.05i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.351 + 2.44i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (2.38 + 1.53i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (4.34 - 9.50i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.37 - 9.57i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (3.64 - 2.34i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 4.14T + 47T^{2} \) |
| 53 | \( 1 + (-10.0 + 2.95i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.31 - 0.679i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (7.25 + 4.66i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (3.11 - 3.59i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (3.79 - 4.37i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.106 - 0.739i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-9.81 - 2.88i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-4.68 + 10.2i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-10.3 + 6.62i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.42 - 5.31i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−9.734721608072259810819985463195, −8.963310598296540755136198301719, −8.149594001466199708655095426453, −7.17231755551905921835242077992, −6.37446751031505960924906820834, −4.89058426089336495001556679688, −4.57409022671355089688831860182, −3.17804080363027339338920845298, −2.25703916588225400811975245183, −0.860703921587916295815984395579,
1.92265920586159050250931138283, 3.33942835366693742011992869599, 3.93218595088316587581397478488, 5.14039040871887134587985291924, 5.90637821217749872427871020130, 7.09398762123701387917012573397, 7.61952891246847076819288611938, 8.645689406406011775584292681051, 9.207573457333072410255911466796, 10.40402286861762680158993333130
