| L(s) = 1 | + (10.5 + 3.96i)2-s + (−64.5 + 26.7i)3-s + (96.5 + 84.0i)4-s + (−11.3 + 27.2i)5-s + (−790. + 27.2i)6-s + (−730. − 730. i)7-s + (689. + 1.27e3i)8-s + (1.90e3 − 1.90e3i)9-s + (−228. + 244. i)10-s + (−4.61e3 − 1.91e3i)11-s + (−8.48e3 − 2.84e3i)12-s + (−4.63e3 − 1.11e4i)13-s + (−4.84e3 − 1.06e4i)14-s − 2.06e3i·15-s + (2.25e3 + 1.62e4i)16-s + 2.43e4i·17-s + ⋯ |
| L(s) = 1 | + (0.936 + 0.350i)2-s + (−1.38 + 0.571i)3-s + (0.754 + 0.656i)4-s + (−0.0404 + 0.0976i)5-s + (−1.49 + 0.0514i)6-s + (−0.805 − 0.805i)7-s + (0.475 + 0.879i)8-s + (0.872 − 0.872i)9-s + (−0.0721 + 0.0772i)10-s + (−1.04 − 0.432i)11-s + (−1.41 − 0.475i)12-s + (−0.584 − 1.41i)13-s + (−0.471 − 1.03i)14-s − 0.157i·15-s + (0.137 + 0.990i)16-s + 1.20i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.689i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0416076 - 0.103975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0416076 - 0.103975i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-10.5 - 3.96i)T \) |
| good | 3 | \( 1 + (64.5 - 26.7i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (11.3 - 27.2i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (730. + 730. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (4.61e3 + 1.91e3i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (4.63e3 + 1.11e4i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 2.43e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (6.49e3 + 1.56e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (5.52e4 - 5.52e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (7.86e4 - 3.25e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.06e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (1.06e5 - 2.56e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-3.32e5 + 3.32e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (1.86e5 + 7.71e4i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 4.75e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (1.47e6 + 6.09e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (7.86e5 - 1.89e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-1.04e6 + 4.31e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-1.19e5 + 4.96e4i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (2.75e6 + 2.75e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (-3.77e6 + 3.77e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 + 2.84e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (4.41e5 + 1.06e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-5.80e6 - 5.80e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 + 1.53e7T + 8.07e13T^{2} \) |
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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
−15.97899330973005357705813493180, −15.12541270657466052941719758678, −13.31165677392201981845447607311, −12.51859495632771574373725688157, −10.96961979742533357719694369289, −10.32284841638439681986088735274, −7.61245542543538795462048477022, −6.10563075839836536072082233153, −5.11357217627114097472921264180, −3.46835756233914867153094387104,
0.04225211393679049020467064630, 2.28455180847542872625844585478, 4.77289309674241954488075977376, 5.98325118358623619728734241089, 7.04823151201641278673160537719, 9.778186228844039184526159586981, 11.20198869314155930220318846938, 12.26490996237357492305946774266, 12.73378212566043831756999685904, 14.29781296024383351883992307400
