| L(s) = 1 | + 2.16i·2-s − 2.56i·3-s + 27.2·4-s + 5.57·6-s + 75.5i·7-s + 128. i·8-s + 236.·9-s + 624.·11-s − 70.1i·12-s − 169i·13-s − 163.·14-s + 594.·16-s − 2.34e3i·17-s + 512. i·18-s + 283.·19-s + ⋯ |
| L(s) = 1 | + 0.383i·2-s − 0.164i·3-s + 0.853·4-s + 0.0631·6-s + 0.583i·7-s + 0.710i·8-s + 0.972·9-s + 1.55·11-s − 0.140i·12-s − 0.277i·13-s − 0.223·14-s + 0.580·16-s − 1.96i·17-s + 0.372i·18-s + 0.180·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.334247928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.334247928\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 2 | \( 1 - 2.16iT - 32T^{2} \) |
| 3 | \( 1 + 2.56iT - 243T^{2} \) |
| 7 | \( 1 - 75.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 624.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 2.34e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 283.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.04e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 687.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.79e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 8.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.32e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.54e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 9.60e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.59e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.54e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.60e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.39e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.01e3iT - 8.58e9T^{2} \) |
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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
−11.08031277700771620970481198719, −9.707209800225198220755367452589, −9.081124816873889724971641964821, −7.62927986632082208265675385195, −7.06413692121648539715785665848, −6.13354678865165297277288440490, −5.05272226834863098845138913528, −3.59005226691299749399250376579, −2.24389808082366553954672723115, −1.09635142873937471557404468002,
1.11956508238631164664216364784, 1.91194990697267016358651702744, 3.68698116314434204709597803059, 4.19219122233422467571212945588, 6.07637490598718414916003825019, 6.78555559797945081866672712560, 7.68367465747946515256152065341, 9.019578055159362458442436782815, 10.01496731917669498611712319077, 10.70405250720178784298019422753
