| L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (1.33 − 2.92i)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 + (3.08 − 0.905i)10-s + (0.342 − 0.395i)11-s + (0.654 − 0.755i)12-s + (2.26 − 0.665i)13-s + (−0.415 − 0.909i)14-s + (−2.70 + 1.73i)15-s + (−0.959 − 0.281i)16-s + (−0.245 − 1.70i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.597 − 1.30i)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 + (0.975 − 0.286i)10-s + (0.103 − 0.119i)11-s + (0.189 − 0.218i)12-s + (0.628 − 0.184i)13-s + (−0.111 − 0.243i)14-s + (−0.698 + 0.448i)15-s + (−0.239 − 0.0704i)16-s + (−0.0595 − 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38604 - 0.862640i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38604 - 0.862640i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.43 + 4.57i)T \) |
| good | 5 | \( 1 + (-1.33 + 2.92i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (-0.342 + 0.395i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.26 + 0.665i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.245 + 1.70i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.352 - 2.44i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (1.42 + 9.91i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.758 + 0.487i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (4.75 + 10.4i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.284 + 0.623i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-3.90 - 2.50i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + (5.93 + 1.74i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (8.73 - 2.56i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.84 + 1.82i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-2.26 - 2.60i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-2.29 - 2.64i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.628 - 4.37i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (6.83 - 2.00i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.04 - 2.28i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-11.7 - 7.53i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (0.0992 - 0.217i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−9.718073055014907783241493237920, −8.931673038064976156618710106907, −8.172091872286550427164699822395, −7.25267677958824403623926445572, −6.04372111962125325215636750661, −5.80480388983783430433622039149, −4.71556263790728047025971435535, −3.89395276698015213003915820784, −2.22424635114927333657324021649, −0.70633042776288307682152212120,
1.65497225412507060347797213922, 2.96653114129412250780611992339, 3.66989609067802434428217457355, 4.92298320024290498923863127044, 5.92435934014393947220043616796, 6.51243286703059223667160823516, 7.30549450269272672659123998012, 8.837993731454088967375731538666, 9.598187653685298430283923039432, 10.51193445512445949547884364614
