| L(s) = 1 | + (−0.887 − 1.10i)2-s + (0.283 − 0.246i)3-s + (−0.426 + 1.95i)4-s + (−2.45 + 0.352i)5-s + (−0.522 − 0.0945i)6-s + (−1.80 + 3.94i)7-s + (2.53 − 1.26i)8-s + (−0.406 + 2.82i)9-s + (2.56 + 2.38i)10-s + (0.456 − 1.55i)11-s + (0.359 + 0.659i)12-s + (−1.85 + 0.845i)13-s + (5.94 − 1.51i)14-s + (−0.609 + 0.703i)15-s + (−3.63 − 1.66i)16-s + (−1.97 − 1.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.627 − 0.778i)2-s + (0.163 − 0.142i)3-s + (−0.213 + 0.977i)4-s + (−1.09 + 0.157i)5-s + (−0.213 − 0.0385i)6-s + (−0.681 + 1.49i)7-s + (0.894 − 0.446i)8-s + (−0.135 + 0.943i)9-s + (0.811 + 0.755i)10-s + (0.137 − 0.468i)11-s + (0.103 + 0.190i)12-s + (−0.513 + 0.234i)13-s + (1.58 − 0.405i)14-s + (−0.157 + 0.181i)15-s + (−0.909 − 0.416i)16-s + (−0.478 − 0.307i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.364053 + 0.293927i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364053 + 0.293927i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.887 + 1.10i)T \) |
| 23 | \( 1 + (-4.47 - 1.71i)T \) |
| good | 3 | \( 1 + (-0.283 + 0.246i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (2.45 - 0.352i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.80 - 3.94i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.456 + 1.55i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.85 - 0.845i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (1.97 + 1.26i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.17 - 1.82i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (2.85 - 4.43i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.79 + 4.37i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (5.95 + 0.856i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.0399 - 0.277i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (3.34 - 2.89i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + (11.4 + 5.23i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-10.7 + 4.91i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-10.0 - 8.70i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.29 - 14.6i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.37 + 0.697i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.91 + 5.73i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.23 - 2.70i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.47 - 0.212i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.94 - 6.86i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.71 - 11.9i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.55718487786160540015284447345, −11.68182332353696926932910848971, −11.13941035848409425590464927830, −9.763789045704322884887245391563, −8.809727972535040372234242766766, −8.050705591069063412649618398962, −6.95091718728387259879361445433, −5.15570770299149408348244204542, −3.49624252690720704683859777517, −2.39357727378778592608357821356,
0.50078962055816155235280044759, 3.65421168150406567692576375408, 4.71145524096535182796229216176, 6.60472966818306576812678984949, 7.19045768132191039213005053470, 8.218995757320653058947847181535, 9.356613470850705945585645680223, 10.18801789451992339892582589920, 11.19884307699863578197515457263, 12.43734675009297876887580701125
