| L(s) = 1 | + 500·25-s + 572·49-s + 4.76e3·73-s + 5.32e3·97-s + 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.01e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4·25-s + 1.66·49-s + 7.63·73-s + 5.56·97-s + 4·121-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 + 0.460·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.548819598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.548819598\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 286 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2}( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 10582 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 35282 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 110 T + p^{3} T^{2} )^{2}( 1 + 110 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 111386 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2}( 1 + 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 172874 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 1190 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 204622 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1330 T + p^{3} T^{2} )^{4} \) |
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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
−7.20299662392624029509997116764, −6.96998242271999813662195366304, −6.70807045534843305589730795178, −6.65586898992746637698457271351, −6.63281908547253912362658800859, −6.05632386883156951524886598175, −5.76527379912107307897262210975, −5.72967424496039947151949618367, −5.27128099269900918543880838778, −4.87141050568116483246671147657, −4.85642264996169452708530555221, −4.76443069749800868349116880033, −4.43037369039784238661009507917, −3.95357947718611103681208369815, −3.44385088941095207008093196323, −3.41849416572924098496586837432, −3.41093966499436541116066998530, −2.81779764860771217270719786386, −2.30206922030250597574865332382, −2.24831316230899120951329007367, −2.07427991504094383689732004954, −1.27866218438550636600268532293, −0.871785814410549917259442415044, −0.794340645454267792128082113077, −0.49048782530309787635686146665,
0.49048782530309787635686146665, 0.794340645454267792128082113077, 0.871785814410549917259442415044, 1.27866218438550636600268532293, 2.07427991504094383689732004954, 2.24831316230899120951329007367, 2.30206922030250597574865332382, 2.81779764860771217270719786386, 3.41093966499436541116066998530, 3.41849416572924098496586837432, 3.44385088941095207008093196323, 3.95357947718611103681208369815, 4.43037369039784238661009507917, 4.76443069749800868349116880033, 4.85642264996169452708530555221, 4.87141050568116483246671147657, 5.27128099269900918543880838778, 5.72967424496039947151949618367, 5.76527379912107307897262210975, 6.05632386883156951524886598175, 6.63281908547253912362658800859, 6.65586898992746637698457271351, 6.70807045534843305589730795178, 6.96998242271999813662195366304, 7.20299662392624029509997116764
