| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (−0.977 + 0.212i)3-s + (−0.989 − 0.142i)4-s + (−0.0118 + 2.23i)5-s + (−0.142 − 0.989i)6-s + (1.21 − 0.663i)7-s + (0.212 − 0.977i)8-s + (0.909 − 0.415i)9-s + (−2.22 − 0.171i)10-s + (3.43 + 2.97i)11-s + (0.997 − 0.0713i)12-s + (−0.0572 + 0.104i)13-s + (0.574 + 1.25i)14-s + (−0.463 − 2.18i)15-s + (0.959 + 0.281i)16-s + (2.58 + 1.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (−0.564 + 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.00529 + 0.999i)5-s + (−0.0580 − 0.404i)6-s + (0.459 − 0.250i)7-s + (0.0751 − 0.345i)8-s + (0.303 − 0.138i)9-s + (−0.705 − 0.0541i)10-s + (1.03 + 0.897i)11-s + (0.287 − 0.0205i)12-s + (−0.0158 + 0.0290i)13-s + (0.153 + 0.336i)14-s + (−0.119 − 0.564i)15-s + (0.239 + 0.0704i)16-s + (0.626 + 0.469i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.331136 + 1.06011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.331136 + 1.06011i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 3 | \( 1 + (0.977 - 0.212i)T \) |
| 5 | \( 1 + (0.0118 - 2.23i)T \) |
| 23 | \( 1 + (4.43 - 1.83i)T \) |
| good | 7 | \( 1 + (-1.21 + 0.663i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-3.43 - 2.97i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.0572 - 0.104i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-2.58 - 1.93i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.165 - 1.15i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-2.45 + 0.352i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (5.49 - 3.52i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.721 - 0.269i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (1.09 - 2.40i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.03 + 4.74i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (3.35 + 3.35i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.37 - 4.35i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (0.267 + 0.912i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-6.89 - 10.7i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (14.4 + 1.03i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-5.17 - 5.96i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (6.67 + 8.91i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 3.31i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.511 + 1.37i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-7.15 - 4.59i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.90 - 13.1i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.59950265739852612826425327854, −10.07005240926852826787409184672, −9.151438303264954256629131607258, −7.923786797080398158159254852757, −7.20458395130177828955822591241, −6.45755776422316795733174688827, −5.63579890949928847282784992882, −4.44477748917135539880929158159, −3.58236190278400727649871153129, −1.68179489591330909608170576485,
0.68899763399899170138065285488, 1.86626903137632956100836654935, 3.55474445497577344527778722143, 4.58162583543244287163995785941, 5.44769423603255194771380935343, 6.32842457034891563375144325817, 7.74990388385831246796389460261, 8.564217597773326284447083394733, 9.304916863777630517743702528214, 10.13397434382190376723454622485
