L(s) = 1 | + 5.65i·2-s − 6.65i·3-s − 32.0·4-s + 146.·5-s + 37.6·6-s − 183.·7-s − 181. i·8-s + 684.·9-s + 831. i·10-s + 1.71e3·11-s + 213. i·12-s + 4.16e3i·13-s − 1.03e3i·14-s − 978. i·15-s + 1.02e3·16-s − 5.34e3·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.246i·3-s − 0.500·4-s + 1.17·5-s + 0.174·6-s − 0.535·7-s − 0.353i·8-s + 0.939·9-s + 0.831i·10-s + 1.29·11-s + 0.123i·12-s + 1.89i·13-s − 0.378i·14-s − 0.289i·15-s + 0.250·16-s − 1.08·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.75635 + 0.963240i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75635 + 0.963240i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 5.65iT \) |
| 19 | \( 1 + (-5.78e3 - 3.68e3i)T \) |
good | 3 | \( 1 + 6.65iT - 729T^{2} \) |
| 5 | \( 1 - 146.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 183.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.71e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 4.16e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 5.34e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 1.74e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 2.12e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 5.17e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 8.13e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 9.84e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.55e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.72e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 3.96e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.93e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 8.06e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.51e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 9.89e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.25e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.37e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.42e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 3.79e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.29e5iT - 8.32e11T^{2} \) |
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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
−15.20630220472490330862752589081, −13.87172095230976500643103970893, −13.30733280262766327429204958065, −11.73887150526391241147029907530, −9.729464475531727766922319194803, −9.143373699046579313091231955265, −6.96674244493476097729150581914, −6.26619633180208426215869370360, −4.31739039182338991389965632526, −1.63320147756624966373125694287,
1.25344850822753627618803157842, 3.18887388143159736315617136492, 5.11314620554732559185490469663, 6.77840863947209273012595571164, 9.031889085593824357853059719902, 9.886400997675688247568086639234, 10.92991800527366117579677416181, 12.70010764676545634319516711416, 13.30380128971167360699652821506, 14.68566924252724091665063218262