| L(s) = 1 | + (3.14 − 2.28i)2-s + (−34.8 + 107. i)4-s + (−310. + 426. i)5-s + (354. + 115. i)7-s + (289. + 890. i)8-s + 2.04e3i·10-s + (−2.32e3 + 3.75e3i)11-s + (2.52e3 + 3.47e3i)13-s + (1.37e3 − 447. i)14-s + (−8.74e3 − 6.35e3i)16-s + (2.43e4 + 1.76e4i)17-s + (−2.20e4 + 7.17e3i)19-s + (−3.50e4 − 4.81e4i)20-s + (1.24e3 + 1.71e4i)22-s − 8.65e4i·23-s + ⋯ |
| L(s) = 1 | + (0.277 − 0.201i)2-s + (−0.272 + 0.838i)4-s + (−1.10 + 1.52i)5-s + (0.390 + 0.127i)7-s + (0.199 + 0.614i)8-s + 0.648i·10-s + (−0.527 + 0.849i)11-s + (0.318 + 0.438i)13-s + (0.134 − 0.0436i)14-s + (−0.534 − 0.388i)16-s + (1.20 + 0.873i)17-s + (−0.738 + 0.239i)19-s + (−0.978 − 1.34i)20-s + (0.0249 + 0.342i)22-s − 1.48i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.235226 - 0.983967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.235226 - 0.983967i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (2.32e3 - 3.75e3i)T \) |
| good | 2 | \( 1 + (-3.14 + 2.28i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (310. - 426. i)T + (-2.41e4 - 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-354. - 115. i)T + (6.66e5 + 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-2.52e3 - 3.47e3i)T + (-1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-2.43e4 - 1.76e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (2.20e4 - 7.17e3i)T + (7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 + 8.65e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + (2.73e4 - 8.41e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-7.97e4 + 5.79e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.06e4 + 3.29e4i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-2.15e5 - 6.61e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 1.81e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + (3.43e5 - 1.11e5i)T + (4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-4.53e5 - 6.24e5i)T + (-3.63e11 + 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-1.97e4 - 6.42e3i)T + (2.01e12 + 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-1.36e6 + 1.87e6i)T + (-9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + 1.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.42e6 - 1.96e6i)T + (-2.81e12 - 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-6.09e6 - 1.98e6i)T + (8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (1.35e6 + 1.87e6i)T + (-5.93e12 + 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-2.33e6 - 1.69e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 1.13e7iT - 4.42e13T^{2} \) |
| 97 | \( 1 + (-2.89e6 + 2.10e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.85799761732423244936397755266, −12.04541097640071732090719493115, −11.12425300340359169823123103046, −10.23146708068718819586790016199, −8.321690004978499065244403263096, −7.68398002086483767408020735798, −6.54429612236941164310343913676, −4.51894449546694219036963281691, −3.53803328755882332184265426236, −2.38696200318356324038665022664,
0.33105443905407986164440792092, 1.16455762170677583970815813150, 3.72071398920536287173196384112, 4.93838731114060412314038219308, 5.64549198302359429496060988507, 7.59048762068265733475105682851, 8.488466544470574120665357168631, 9.567615901722535656794272394546, 10.96502655783165571447438632052, 11.94685999805077895745712736984
