| L(s) = 1 | + (0.959 − 0.281i)2-s − 3-s + (0.841 − 0.540i)4-s + (2.76 + 3.18i)5-s + (−0.959 + 0.281i)6-s + (−0.612 − 1.34i)7-s + (0.654 − 0.755i)8-s + 9-s + (3.54 + 2.27i)10-s + (2.85 − 1.69i)11-s + (−0.841 + 0.540i)12-s + (−1.32 + 0.852i)13-s + (−0.966 − 1.11i)14-s + (−2.76 − 3.18i)15-s + (0.415 − 0.909i)16-s + (−0.0558 + 0.388i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s − 0.577·3-s + (0.420 − 0.270i)4-s + (1.23 + 1.42i)5-s + (−0.391 + 0.115i)6-s + (−0.231 − 0.507i)7-s + (0.231 − 0.267i)8-s + 0.333·9-s + (1.12 + 0.720i)10-s + (0.859 − 0.510i)11-s + (−0.242 + 0.156i)12-s + (−0.367 + 0.236i)13-s + (−0.258 − 0.297i)14-s + (−0.712 − 0.822i)15-s + (0.103 − 0.227i)16-s + (−0.0135 + 0.0942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.33438 + 0.375606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.33438 + 0.375606i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + (-2.85 + 1.69i)T \) |
| good | 5 | \( 1 + (-2.76 - 3.18i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.612 + 1.34i)T + (-4.58 + 5.29i)T^{2} \) |
| 13 | \( 1 + (1.32 - 0.852i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.0558 - 0.388i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.581 - 4.04i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (-2.07 + 4.54i)T + (-15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 + (-0.724 - 5.04i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-4.06 - 2.60i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (3.33 + 2.14i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (1.78 - 0.523i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-2.01 + 2.32i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-7.15 - 2.10i)T + (39.5 + 25.4i)T^{2} \) |
| 53 | \( 1 + (3.67 + 8.03i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (9.60 + 2.82i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (4.75 + 1.39i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.21 + 0.357i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-0.922 - 6.41i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (0.136 - 0.297i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (7.01 + 8.09i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (5.29 + 11.5i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (2.42 - 16.8i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.44 + 7.43i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.47241195301564060108581211496, −10.05310381112207250807221016328, −8.991897236101844500070088244035, −7.33058729090672080158398270264, −6.60056736428230942050914363266, −6.15493191913498193667230109869, −5.16760971824335554327802470571, −3.83456717141187452810433360761, −2.88107951024984218859551371445, −1.59116041411549282765127020553,
1.26821596998215297828921795824, 2.53766307350591880153793742022, 4.31896799755506284316619994411, 5.00813635035694761015899517578, 5.78952078275974340305640954163, 6.43486373554503500518235349166, 7.59158704486821127728278758890, 8.923448665425060771414638456585, 9.396255236204439480387320825013, 10.23415664976565694646750769831
