| L(s) = 1 | + 4·4-s + 6·9-s − 12·11-s + 4·16-s + 24·19-s + 12·29-s − 24·31-s + 24·36-s + 8·41-s − 48·44-s + 8·59-s − 16·64-s − 24·71-s + 96·76-s + 28·79-s + 17·81-s − 32·89-s − 72·99-s + 24·101-s − 20·109-s + 48·116-s + 62·121-s − 96·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2·4-s + 2·9-s − 3.61·11-s + 16-s + 5.50·19-s + 2.22·29-s − 4.31·31-s + 4·36-s + 1.24·41-s − 7.23·44-s + 1.04·59-s − 2·64-s − 2.84·71-s + 11.0·76-s + 3.15·79-s + 17/9·81-s − 3.39·89-s − 7.23·99-s + 2.38·101-s − 1.91·109-s + 4.45·116-s + 5.63·121-s − 8.62·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.522267366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.522267366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 491 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 43 p T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 834 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 8259 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 4 T + 104 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 17826 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_4$ | \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 22011 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.98796468974389502323412828151, −6.98363290492746593740983918365, −6.84045061439501602714711681239, −6.34696800522362573593715694532, −5.84192138597272278981041339639, −5.78928336388142164310111430713, −5.64471692916712948508154018816, −5.33421194560388637154149162104, −5.32645511438736651570010808757, −4.97473790733920081220180753826, −4.87825936341457816741149423533, −4.53872707093786199692961572561, −4.10721730909854452609510069331, −4.02768448642237215175657012170, −3.43116280685049105229478629439, −3.25529125406817869078892988610, −3.00045444112875943221989240551, −2.94969784261575023663820918899, −2.68184558177200711914750519250, −2.28259666147370226184679726090, −2.01953311062371501653911428773, −1.65146466393061523194375995608, −1.44714350332016053173373185346, −0.959370353630057669573196307846, −0.50832121335449841800534236296,
0.50832121335449841800534236296, 0.959370353630057669573196307846, 1.44714350332016053173373185346, 1.65146466393061523194375995608, 2.01953311062371501653911428773, 2.28259666147370226184679726090, 2.68184558177200711914750519250, 2.94969784261575023663820918899, 3.00045444112875943221989240551, 3.25529125406817869078892988610, 3.43116280685049105229478629439, 4.02768448642237215175657012170, 4.10721730909854452609510069331, 4.53872707093786199692961572561, 4.87825936341457816741149423533, 4.97473790733920081220180753826, 5.32645511438736651570010808757, 5.33421194560388637154149162104, 5.64471692916712948508154018816, 5.78928336388142164310111430713, 5.84192138597272278981041339639, 6.34696800522362573593715694532, 6.84045061439501602714711681239, 6.98363290492746593740983918365, 6.98796468974389502323412828151
