| L(s) = 1 | + 42·9-s − 104·13-s − 76·25-s + 824·37-s + 772·49-s + 1.11e3·61-s − 1.68e3·73-s + 1.03e3·81-s − 4.28e3·97-s + 1.43e3·109-s − 4.36e3·117-s − 3.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.02e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 14/9·9-s − 2.21·13-s − 0.607·25-s + 3.66·37-s + 2.25·49-s + 2.33·61-s − 2.70·73-s + 1.41·81-s − 4.48·97-s + 1.25·109-s − 3.45·117-s − 2.70·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.923·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.028356504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028356504\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 14 p T^{2} + p^{6} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 386 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 1798 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 5218 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2186 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 6770 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 48490 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 58610 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 206 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 44530 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 150266 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 193822 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 295162 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 98854 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 278 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 191062 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 712366 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 422 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 539090 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 1142710 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 1270546 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1070 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.28145290676356599116562575691, −10.92304465276536632813241087427, −10.38761988252446993438826674431, −10.16118561804820831902803443080, −9.832731730861270285636002974219, −9.773743803575971176829387282759, −9.462328934448477676119150971924, −8.848649501376301939389858614993, −8.807879861310938851042297872416, −7.981597473804520481216973455153, −7.69356401754218641905827460137, −7.65139944315157107934638018104, −7.19896939401082726898044586355, −6.77038550770534651089334425737, −6.56625715956571960629456174105, −5.92053759936660615603833744957, −5.35544127540697472462002138247, −5.28333118641000321052146491692, −4.39581365844079065872555497017, −4.17134027108114254946896496833, −4.10409578895279394935748584838, −2.72010089401387602088294765869, −2.67978124865639315022734160496, −1.75494584468707602712319511577, −0.71455361788510868503819984830,
0.71455361788510868503819984830, 1.75494584468707602712319511577, 2.67978124865639315022734160496, 2.72010089401387602088294765869, 4.10409578895279394935748584838, 4.17134027108114254946896496833, 4.39581365844079065872555497017, 5.28333118641000321052146491692, 5.35544127540697472462002138247, 5.92053759936660615603833744957, 6.56625715956571960629456174105, 6.77038550770534651089334425737, 7.19896939401082726898044586355, 7.65139944315157107934638018104, 7.69356401754218641905827460137, 7.981597473804520481216973455153, 8.807879861310938851042297872416, 8.848649501376301939389858614993, 9.462328934448477676119150971924, 9.773743803575971176829387282759, 9.832731730861270285636002974219, 10.16118561804820831902803443080, 10.38761988252446993438826674431, 10.92304465276536632813241087427, 11.28145290676356599116562575691
