L(s) = 1 | + (−0.449 − 1.67i)2-s + i·3-s + (−0.885 + 0.511i)4-s + (−0.524 + 1.95i)5-s + (1.67 − 0.449i)6-s + (−0.616 + 2.57i)7-s + (−1.20 − 1.20i)8-s − 9-s + 3.52·10-s + (2.56 + 2.56i)11-s + (−0.511 − 0.885i)12-s + (1.78 + 3.13i)13-s + (4.59 − 0.123i)14-s + (−1.95 − 0.524i)15-s + (−2.49 + 4.32i)16-s + (−0.752 − 1.30i)17-s + ⋯ |
L(s) = 1 | + (−0.318 − 1.18i)2-s + 0.577i·3-s + (−0.442 + 0.255i)4-s + (−0.234 + 0.875i)5-s + (0.685 − 0.183i)6-s + (−0.232 + 0.972i)7-s + (−0.424 − 0.424i)8-s − 0.333·9-s + 1.11·10-s + (0.772 + 0.772i)11-s + (−0.147 − 0.255i)12-s + (0.496 + 0.868i)13-s + (1.22 − 0.0328i)14-s + (−0.505 − 0.135i)15-s + (−0.624 + 1.08i)16-s + (−0.182 − 0.316i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.969391 + 0.188908i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.969391 + 0.188908i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (0.616 - 2.57i)T \) |
| 13 | \( 1 + (-1.78 - 3.13i)T \) |
good | 2 | \( 1 + (0.449 + 1.67i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (0.524 - 1.95i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.56 - 2.56i)T + 11iT^{2} \) |
| 17 | \( 1 + (0.752 + 1.30i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.92 - 2.92i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.21 + 1.85i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.84 + 3.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.05 + 1.89i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (2.52 - 0.677i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.33 + 4.97i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (4.51 + 2.60i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.255 - 0.0684i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (2.63 - 4.57i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.24 - 0.334i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 10.6iT - 61T^{2} \) |
| 67 | \( 1 + (-9.48 + 9.48i)T - 67iT^{2} \) |
| 71 | \( 1 + (-3.11 - 11.6i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.744 + 2.77i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (8.09 + 14.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.3 - 10.3i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.60 + 9.72i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.645 + 0.173i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.81828431502505724387464529013, −11.10125332715105574808030856170, −10.05420268451923574153179089605, −9.496922000224253904713938359675, −8.578417579458423588611181894660, −6.92903267376326334666240172698, −6.00307881404463479033100567192, −4.24509649495812454669450699512, −3.19744428621388232543840758845, −2.03267322909220907518490370854,
0.883303884580761441661715062658, 3.39524890690245044100651432202, 4.98366847189238389982577775682, 6.15771949178571475533378942543, 6.91842377980054855161569263612, 8.016857058462430211238602451340, 8.491634024372078585358894967789, 9.580024402617244368114916646074, 11.00801277688028892762760285591, 11.89918737588625927957776486410