L(s) = 1 | + 4-s + 4·16-s − 16·19-s + 5·25-s − 16·31-s − 14·49-s − 4·61-s + 11·64-s − 16·76-s + 32·79-s + 5·100-s + 56·109-s + 22·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 16-s − 3.67·19-s + 25-s − 2.87·31-s − 2·49-s − 0.512·61-s + 11/8·64-s − 1.83·76-s + 3.60·79-s + 1/2·100-s + 5.36·109-s + 2·121-s − 1.43·124-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 − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.378150229\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378150229\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 14 T^{2} - 2013 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 154 T^{2} + 16827 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.369814538226362840518952268733, −8.004893326395897102334949780281, −7.48613015698205031967928444639, −7.41347174060500359672031205677, −7.20951703150063010986622023724, −6.95436069193290696773683620165, −6.66135345361517135208058077696, −6.27365804962657593005476144460, −6.06176014302803497260194533397, −6.05964597936380471116148728792, −5.88884405147966952323756211855, −5.18010291709208538409372949525, −4.96533041134637229405097271559, −4.86278321638879379767305792393, −4.61365207986937693365452577404, −4.10498306318368326979368505879, −3.74433880585765858894579218498, −3.58117302598872744835150085828, −3.40819861972392036889456005048, −2.87834608732151147481978912402, −2.33042012379504187933894482332, −2.06485102162863990321339208821, −1.95545951701561982223894783375, −1.34507427513411275027406755404, −0.43020214201801468131864267480,
0.43020214201801468131864267480, 1.34507427513411275027406755404, 1.95545951701561982223894783375, 2.06485102162863990321339208821, 2.33042012379504187933894482332, 2.87834608732151147481978912402, 3.40819861972392036889456005048, 3.58117302598872744835150085828, 3.74433880585765858894579218498, 4.10498306318368326979368505879, 4.61365207986937693365452577404, 4.86278321638879379767305792393, 4.96533041134637229405097271559, 5.18010291709208538409372949525, 5.88884405147966952323756211855, 6.05964597936380471116148728792, 6.06176014302803497260194533397, 6.27365804962657593005476144460, 6.66135345361517135208058077696, 6.95436069193290696773683620165, 7.20951703150063010986622023724, 7.41347174060500359672031205677, 7.48613015698205031967928444639, 8.004893326395897102334949780281, 8.369814538226362840518952268733