| L(s) = 1 | + (−4.5 − 7.79i)3-s + (13.3 − 23.1i)5-s + (66.5 + 111. i)7-s + (−40.5 + 70.1i)9-s + (−81.6 − 141. i)11-s − 595.·13-s − 240.·15-s + (301. + 522. i)17-s + (1.24e3 − 2.14e3i)19-s + (567. − 1.01e3i)21-s + (784. − 1.35e3i)23-s + (1.20e3 + 2.08e3i)25-s + 729·27-s − 2.06e3·29-s + (2.99e3 + 5.19e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.239 − 0.414i)5-s + (0.513 + 0.858i)7-s + (−0.166 + 0.288i)9-s + (−0.203 − 0.352i)11-s − 0.977·13-s − 0.276·15-s + (0.253 + 0.438i)17-s + (0.788 − 1.36i)19-s + (0.280 − 0.504i)21-s + (0.309 − 0.535i)23-s + (0.385 + 0.667i)25-s + 0.192·27-s − 0.454·29-s + (0.560 + 0.970i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0791 + 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0791 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.696211721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.696211721\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 + 7.79i)T \) |
| 7 | \( 1 + (-66.5 - 111. i)T \) |
| good | 5 | \( 1 + (-13.3 + 23.1i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (81.6 + 141. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 595.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-301. - 522. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.24e3 + 2.14e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-784. + 1.35e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 2.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-2.99e3 - 5.19e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-295. + 511. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 2.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.41e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.35e4 + 2.34e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.60e4 + 2.77e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (5.17e3 + 8.96e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-5.20e3 + 9.01e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.33e4 + 4.03e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 4.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (9.58e3 + 1.65e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.55e4 + 6.16e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 7.30e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.21e4 + 2.10e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 9.27e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.65180986259342891905682767609, −9.398592026678376159546140388339, −8.662365645916742180510672863672, −7.64687852664199771887459627006, −6.64178838005735741186937303373, −5.38519490028468891790036173591, −4.90144502221539381987148678953, −2.98250575870980891276024796572, −1.85795741382620708447515067772, −0.52441766937056080810310644395,
1.06551999861815974487941751322, 2.62948298995252092047539688386, 3.97243900891378924890722577191, 4.93632523361483974382965411609, 5.97546768276679699994531008619, 7.27931669432967498702274143349, 7.86620608351271528913850270716, 9.423854364374260213779253750091, 10.06540954745734755494361980928, 10.80053862863849628904492775849
