| L(s) = 1 | + 68·9-s − 184·17-s − 50·25-s + 40·41-s + 588·49-s − 2.08e3·73-s + 2.01e3·81-s − 3.88e3·89-s − 3.73e3·97-s − 8.72e3·113-s + 5.16e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.25e4·153-s + 157-s + 163-s + 167-s + 1.10e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2.51·9-s − 2.62·17-s − 2/5·25-s + 0.152·41-s + 12/7·49-s − 3.34·73-s + 2.75·81-s − 4.62·89-s − 3.91·97-s − 7.26·113-s + 3.87·121-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 − 6.61·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.500·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6576022627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6576022627\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 p^{2} T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2582 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 550 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 46 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2198 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 12646 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40678 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 36462 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )^{2}( 1 + 396 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 154514 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 49226 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 161930 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 241478 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 391462 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 599106 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 581342 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 522 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 217758 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 999074 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 970 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 934 T + p^{3} T^{2} )^{4} \) |
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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
−6.66002748437146402496611828825, −6.65356687791172360875548144603, −6.04600888502554303506236424203, −5.68629735062512243416965916287, −5.60618084651259818025515095374, −5.55217778415788084856059570988, −5.31999676510117147675401889573, −4.64940045137544488462725649025, −4.52706114123846418432775803782, −4.44457021036536576879944503336, −4.37893453975748396549307363199, −4.05262421936386100360940474467, −3.87136145907700723040194633779, −3.79535420516868782820266454424, −3.01918008628723528847185945080, −2.95669779110963077305929985737, −2.77270845465095908462591800105, −2.38715769650539617571227666553, −2.05826005077091904047821762205, −1.79226970452170293497860322022, −1.50222520312761604120506180272, −1.34829455917969117263002784972, −1.02433737059245267682862169867, −0.48551015029238458257720659547, −0.10031340865744941966204890476,
0.10031340865744941966204890476, 0.48551015029238458257720659547, 1.02433737059245267682862169867, 1.34829455917969117263002784972, 1.50222520312761604120506180272, 1.79226970452170293497860322022, 2.05826005077091904047821762205, 2.38715769650539617571227666553, 2.77270845465095908462591800105, 2.95669779110963077305929985737, 3.01918008628723528847185945080, 3.79535420516868782820266454424, 3.87136145907700723040194633779, 4.05262421936386100360940474467, 4.37893453975748396549307363199, 4.44457021036536576879944503336, 4.52706114123846418432775803782, 4.64940045137544488462725649025, 5.31999676510117147675401889573, 5.55217778415788084856059570988, 5.60618084651259818025515095374, 5.68629735062512243416965916287, 6.04600888502554303506236424203, 6.65356687791172360875548144603, 6.66002748437146402496611828825
