| L(s) = 1 | + 8·17-s + 4·25-s − 40·41-s − 12·49-s − 8·73-s − 24·89-s + 56·97-s − 8·113-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.94·17-s + 4/5·25-s − 6.24·41-s − 1.71·49-s − 0.936·73-s − 2.54·89-s + 5.68·97-s − 0.752·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8221007509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8221007509\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
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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
−7.00150158332831843107301464615, −6.70989442004737162871460805361, −6.51534251112862772691056997974, −6.47923965040494066937806798261, −6.07217901468720812716646738744, −6.06172355475996944645834051201, −5.56922694993706530252994530735, −5.29983294506024541734843904276, −5.16977221101405703143727365532, −5.07270764464667137447942657952, −4.75629565840667196882553723715, −4.73583066904685168062301457370, −4.12678273599902205904955052875, −3.93630362445975200210690591408, −3.71908674026926067478169834057, −3.20577265583575093578349322230, −3.18874602338073949212864779184, −3.18376368063258798649911542502, −2.84198226003781938157387398238, −2.08086361289812142499447582937, −2.02931349115505972846574068160, −1.66601380241701012587719386530, −1.28320948559777165490831403198, −1.02599418537207740028973219272, −0.19731264246088951185321303071,
0.19731264246088951185321303071, 1.02599418537207740028973219272, 1.28320948559777165490831403198, 1.66601380241701012587719386530, 2.02931349115505972846574068160, 2.08086361289812142499447582937, 2.84198226003781938157387398238, 3.18376368063258798649911542502, 3.18874602338073949212864779184, 3.20577265583575093578349322230, 3.71908674026926067478169834057, 3.93630362445975200210690591408, 4.12678273599902205904955052875, 4.73583066904685168062301457370, 4.75629565840667196882553723715, 5.07270764464667137447942657952, 5.16977221101405703143727365532, 5.29983294506024541734843904276, 5.56922694993706530252994530735, 6.06172355475996944645834051201, 6.07217901468720812716646738744, 6.47923965040494066937806798261, 6.51534251112862772691056997974, 6.70989442004737162871460805361, 7.00150158332831843107301464615
