| L(s) = 1 | + (−11.2 − 1.29i)2-s + (−18.4 + 44.5i)3-s + (124. + 29.1i)4-s + (118. − 49.0i)5-s + (265. − 476. i)6-s + (519. − 519. i)7-s + (−1.36e3 − 490. i)8-s + (−98.6 − 98.6i)9-s + (−1.39e3 + 397. i)10-s + (323. + 781. i)11-s + (−3.60e3 + 5.01e3i)12-s + (5.06e3 + 2.09e3i)13-s + (−6.50e3 + 5.16e3i)14-s + 6.18e3i·15-s + (1.46e4 + 7.27e3i)16-s + 3.97e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.993 − 0.114i)2-s + (−0.394 + 0.952i)3-s + (0.973 + 0.228i)4-s + (0.423 − 0.175i)5-s + (0.501 − 0.901i)6-s + (0.572 − 0.572i)7-s + (−0.941 − 0.338i)8-s + (−0.0450 − 0.0450i)9-s + (−0.441 + 0.125i)10-s + (0.0733 + 0.177i)11-s + (−0.601 + 0.837i)12-s + (0.638 + 0.264i)13-s + (−0.633 + 0.502i)14-s + 0.473i·15-s + (0.895 + 0.444i)16-s + 1.96i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0137 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.718444 + 0.728378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.718444 + 0.728378i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.2 + 1.29i)T \) |
| good | 3 | \( 1 + (18.4 - 44.5i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-118. + 49.0i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-519. + 519. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-323. - 781. i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-5.06e3 - 2.09e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 3.97e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (1.73e4 + 7.20e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-3.68e3 - 3.68e3i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (1.08e4 - 2.62e4i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 5.83e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-3.54e5 + 1.46e5i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-2.26e5 - 2.26e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-6.24e4 - 1.50e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 6.95e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-2.66e5 - 6.42e5i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (7.16e5 - 2.96e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-6.77e5 + 1.63e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-9.70e5 + 2.34e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (1.48e6 - 1.48e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-3.76e6 - 3.76e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 6.70e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (5.73e6 + 2.37e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-8.68e6 + 8.68e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 5.04e4T + 8.07e13T^{2} \) |
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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
−15.86204177607609154535266403439, −14.77799811558976857312891726831, −12.93917765589155653747564472886, −11.16380829598941311315907113854, −10.51707339167160313653201617121, −9.337105877479007968234520909076, −7.925594115439944968446267889012, −6.08020863235525874417306116089, −4.12558188515316827526803523076, −1.55553839684648875793672834562,
0.74281222948113569477665976775, 2.27243654034067825490110690657, 5.76948725777437463439787307100, 6.95354055425853634650951900434, 8.274222609153259758077948288447, 9.685343268468436093118758218910, 11.25845665369635434671457598707, 12.13168296024426190385087311887, 13.69644479414800503018684479597, 15.17684829015272433094677916117
