| L(s) = 1 | + (−367. + 367. i)3-s + (−502. − 502. i)5-s + 3.30e4i·7-s − 9.34e4i·9-s + (−3.35e5 − 3.35e5i)11-s + (5.68e5 − 5.68e5i)13-s + 3.69e5·15-s − 6.78e6·17-s + (−1.33e6 + 1.33e6i)19-s + (−1.21e7 − 1.21e7i)21-s − 1.07e6i·23-s − 4.83e7i·25-s + (−3.07e7 − 3.07e7i)27-s + (5.01e7 − 5.01e7i)29-s − 2.95e8·31-s + ⋯ |
| L(s) = 1 | + (−0.873 + 0.873i)3-s + (−0.0718 − 0.0718i)5-s + 0.743i·7-s − 0.527i·9-s + (−0.628 − 0.628i)11-s + (0.424 − 0.424i)13-s + 0.125·15-s − 1.15·17-s + (−0.124 + 0.124i)19-s + (−0.649 − 0.649i)21-s − 0.0347i·23-s − 0.989i·25-s + (−0.412 − 0.412i)27-s + (0.453 − 0.453i)29-s − 1.85·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00146i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.930960 + 0.000679901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930960 + 0.000679901i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (367. - 367. i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (502. + 502. i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 - 3.30e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (3.35e5 + 3.35e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (-5.68e5 + 5.68e5i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 + 6.78e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (1.33e6 - 1.33e6i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 + 1.07e6iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-5.01e7 + 5.01e7i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 + 2.95e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-5.11e8 - 5.11e8i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 + 9.31e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-1.27e9 - 1.27e9i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 - 1.80e7T + 2.47e18T^{2} \) |
| 53 | \( 1 + (-2.29e9 - 2.29e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (-4.81e8 - 4.81e8i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (-1.30e9 + 1.30e9i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (-9.42e9 + 9.42e9i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 + 1.74e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 8.91e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 4.07e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (1.16e10 - 1.16e10i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 - 7.96e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 1.03e10T + 7.15e21T^{2} \) |
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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
−12.42959305161199001016478259473, −11.21165615487493185926552396797, −10.60139098214444635096917798301, −9.269112704666819640064285135868, −8.094180839552541355734097593112, −6.21155038064353145257312001265, −5.33912054583979713044055377544, −4.14908683090914467997909610635, −2.48748578852389232808887969071, −0.41756075774334416651514619482,
0.74255949640646098761029052494, 2.04683396858865349221885193423, 4.02589589421680264582187799773, 5.49367466663798092718178622690, 6.80218568180395584168100465255, 7.47329299684369815923642991762, 9.147039440139441614075773834013, 10.70832173205809606879941659830, 11.41264947118885308730728436817, 12.73919042112875923067978767915
