| L(s) = 1 | − 16·4-s + 4·7-s − 30·9-s + 160·16-s − 340·25-s − 64·28-s + 480·36-s − 592·37-s − 1.69e3·43-s + 1.10e3·49-s − 120·63-s − 1.28e3·64-s + 496·67-s + 2.82e3·79-s + 450·81-s + 5.44e3·100-s − 544·109-s + 640·112-s + 4.81e3·121-s + 127-s + 131-s + 137-s + 139-s − 4.80e3·144-s + 9.47e3·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2·4-s + 0.215·7-s − 1.11·9-s + 5/2·16-s − 2.71·25-s − 0.431·28-s + 20/9·36-s − 2.63·37-s − 6.01·43-s + 3.21·49-s − 0.239·63-s − 5/2·64-s + 0.904·67-s + 4.02·79-s + 0.617·81-s + 5.43·100-s − 0.478·109-s + 0.539·112-s + 3.61·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 2.77·144-s + 5.26·148-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.03619651676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03619651676\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 3 | \( 1 + 10 p T^{2} + 50 p^{2} T^{4} + 10 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | \( ( 1 - 2 T - 78 p T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| good | 5 | \( ( 1 + 34 p T^{2} + 37242 T^{4} + 34 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2408 T^{2} + 3883038 T^{4} - 2408 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 5830 T^{2} + 16965930 T^{4} - 5830 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 + 208 p T^{2} + 26538750 T^{4} + 208 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 - 8674 T^{2} + 67920258 T^{4} - 8674 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 24692 T^{2} + 309294726 T^{4} - 24692 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 77792 T^{2} + 2659888590 T^{4} - 77792 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 - 6304 T^{2} - 228896706 T^{4} - 6304 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 p T + p^{3} T^{2} )^{8} \) |
| 41 | \( ( 1 + 180944 T^{2} + 15451545438 T^{4} + 180944 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 + 424 T + 125046 T^{2} + 424 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 47 | \( ( 1 + 159812 T^{2} + 21556242486 T^{4} + 159812 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 386240 T^{2} + 79722075246 T^{4} - 386240 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 + 240686 T^{2} + 18524601378 T^{4} + 240686 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 724246 T^{2} + 234171743178 T^{4} - 724246 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 124 T + 600438 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 71 | \( ( 1 - 823928 T^{2} + 346330727790 T^{4} - 823928 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 - 456388 T^{2} + 201396314982 T^{4} - 456388 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 706 T + 1010814 T^{2} - 706 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 + 784286 T^{2} + 665936055234 T^{4} + 784286 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 + 2427080 T^{2} + 2463477537390 T^{4} + 2427080 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2312332 T^{2} + 2649505779606 T^{4} - 2312332 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.23670168963456508027068314974, −7.12099770454849448244348515699, −6.90353928269277259597319353277, −6.66023306481880708202482098230, −6.52866086256788610997854163857, −6.08838109242882365348290822461, −6.04934962374267104399854755887, −5.87799248022677346526491159094, −5.52026282644537476930465003559, −5.19183307225042634058589201759, −5.10951949173703563443804821941, −5.10111166314181345516756565251, −5.00656090231212589027643228140, −4.45277278122729543350058192679, −4.12428640653525301147765082271, −4.06735620945415211898759578671, −3.58508790093544068832456047632, −3.48510342266078607840559528230, −3.40407889333021564086268756636, −3.01519619078416127378215616457, −2.32043329302651696119709591756, −1.89935550184982815033040953295, −1.81409075161617370822042647845, −0.846162881185845017571493382553, −0.06986784575391022400258742491,
0.06986784575391022400258742491, 0.846162881185845017571493382553, 1.81409075161617370822042647845, 1.89935550184982815033040953295, 2.32043329302651696119709591756, 3.01519619078416127378215616457, 3.40407889333021564086268756636, 3.48510342266078607840559528230, 3.58508790093544068832456047632, 4.06735620945415211898759578671, 4.12428640653525301147765082271, 4.45277278122729543350058192679, 5.00656090231212589027643228140, 5.10111166314181345516756565251, 5.10951949173703563443804821941, 5.19183307225042634058589201759, 5.52026282644537476930465003559, 5.87799248022677346526491159094, 6.04934962374267104399854755887, 6.08838109242882365348290822461, 6.52866086256788610997854163857, 6.66023306481880708202482098230, 6.90353928269277259597319353277, 7.12099770454849448244348515699, 7.23670168963456508027068314974
Plot not available for L-functions of degree greater than 10.
