| L(s) = 1 | + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (−0.142 − 0.989i)6-s + (1.17 + 0.343i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (2.88 − 3.33i)11-s + (0.654 − 0.755i)12-s + (0.223 − 0.0655i)13-s + (0.506 + 1.10i)14-s + (0.841 − 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.904 + 6.29i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (−0.185 + 0.406i)5-s + (−0.0580 − 0.404i)6-s + (0.442 + 0.129i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.303 + 0.0890i)10-s + (0.870 − 1.00i)11-s + (0.189 − 0.218i)12-s + (0.0619 − 0.0181i)13-s + (0.135 + 0.296i)14-s + (0.217 − 0.139i)15-s + (−0.239 − 0.0704i)16-s + (0.219 + 1.52i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21967 + 1.08609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21967 + 1.08609i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.39 - 1.92i)T \) |
| good | 7 | \( 1 + (-1.17 - 0.343i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.88 + 3.33i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.223 + 0.0655i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.904 - 6.29i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.986 - 6.86i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.563 + 3.91i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.83 - 2.46i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.30 - 7.24i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (2.31 - 5.06i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.94 + 1.25i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 9.98T + 47T^{2} \) |
| 53 | \( 1 + (-1.13 - 0.332i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-10.8 + 3.17i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.566 + 0.364i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (2.29 + 2.64i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-1.94 - 2.24i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.990i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (0.718 - 0.210i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.0392 - 0.0858i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (4.63 + 2.97i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-5.63 + 12.3i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.90642873116214495964855399848, −9.905916489700684817991777679047, −8.527553590389174693671507770973, −8.081086602257652457631805442887, −6.95970308507285637936256924068, −6.12169535425501966144209674493, −5.56412170376960962804474031042, −4.18909305017663355980828234691, −3.34979387406387578667814544337, −1.55109665161267533530778133237,
0.884633990006004770099556248895, 2.44867480694683889818752504435, 3.91519152440059505004439085129, 4.75785888032621716061316062046, 5.32750153594277101490651114028, 6.76550334730083573944379675914, 7.37863955552715078158923799037, 9.081688165148351223216272689412, 9.273151154122864453089026626771, 10.49221151680112199919919103952
