| L(s) = 1 | + 6·7-s − 34·13-s + 22·19-s + 18·25-s + 100·31-s − 66·37-s + 20·43-s − 71·49-s − 82·61-s + 166·67-s + 254·73-s + 38·79-s − 204·91-s + 334·97-s − 106·103-s + 20·109-s + 210·121-s + 127-s + 131-s + 132·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 6/7·7-s − 2.61·13-s + 1.15·19-s + 0.719·25-s + 3.22·31-s − 1.78·37-s + 0.465·43-s − 1.44·49-s − 1.34·61-s + 2.47·67-s + 3.47·73-s + 0.481·79-s − 2.24·91-s + 3.44·97-s − 1.02·103-s + 0.183·109-s + 1.73·121-s + 0.00787·127-s + 0.00763·131-s + 0.992·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.894734848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.894734848\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 18 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 222 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 510 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 33 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2210 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2782 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5490 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6162 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 41 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 9570 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 127 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 19 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1710 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 8642 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.14726947026246155504215770669, −12.10814744032847541178956038954, −11.40599647432225203179388334938, −10.95509757886721116734760142887, −10.29223379469338757686925298953, −9.734215851575216606019678866892, −9.713383141607929016613142171665, −8.897651941695175288961062409735, −8.132605755575295443649256126213, −7.958718841447140589960764660766, −7.31507933911520968485166665514, −6.81170480450691788822144344305, −6.28003321720746245590593858284, −5.17754406266611476507523462822, −4.93509616168324422679468120437, −4.66844462646731653008030244918, −3.48488220339928317122219399230, −2.72031815038004948737886606450, −2.05405288164997694335014978967, −0.78174922194648870619651302241,
0.78174922194648870619651302241, 2.05405288164997694335014978967, 2.72031815038004948737886606450, 3.48488220339928317122219399230, 4.66844462646731653008030244918, 4.93509616168324422679468120437, 5.17754406266611476507523462822, 6.28003321720746245590593858284, 6.81170480450691788822144344305, 7.31507933911520968485166665514, 7.958718841447140589960764660766, 8.132605755575295443649256126213, 8.897651941695175288961062409735, 9.713383141607929016613142171665, 9.734215851575216606019678866892, 10.29223379469338757686925298953, 10.95509757886721116734760142887, 11.40599647432225203179388334938, 12.10814744032847541178956038954, 12.14726947026246155504215770669
