| L(s) = 1 | − 6·3-s + 2·5-s + 15·9-s + 3·11-s − 12·15-s + 9·17-s + 19-s + 12·23-s − 7·25-s − 18·27-s − 2·29-s − 31-s − 18·33-s − 27·37-s + 30·41-s + 30·45-s − 15·47-s − 54·51-s + 3·53-s + 6·55-s − 6·57-s − 59-s − 12·61-s + 18·67-s − 72·69-s + 12·71-s − 15·73-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 0.894·5-s + 5·9-s + 0.904·11-s − 3.09·15-s + 2.18·17-s + 0.229·19-s + 2.50·23-s − 7/5·25-s − 3.46·27-s − 0.371·29-s − 0.179·31-s − 3.13·33-s − 4.43·37-s + 4.68·41-s + 4.47·45-s − 2.18·47-s − 7.56·51-s + 0.412·53-s + 0.809·55-s − 0.794·57-s − 0.130·59-s − 1.53·61-s + 2.19·67-s − 8.66·69-s + 1.42·71-s − 1.75·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9290553517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9290553517\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - T^{2} + 36 T^{3} - 120 T^{4} + 36 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 9 T + 63 T^{2} - 324 T^{3} + 1466 T^{4} - 324 p T^{5} + 63 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - T - 23 T^{2} + 14 T^{3} + 196 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_{4}$ | \( ( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + T - 47 T^{2} - 14 T^{3} + 1312 T^{4} - 14 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 27 T + 373 T^{2} + 3510 T^{3} + 24522 T^{4} + 3510 p T^{5} + 373 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 15 T + 124 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 7248 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 15 T + 183 T^{2} + 1620 T^{3} + 12980 T^{4} + 1620 p T^{5} + 183 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3 T + 67 T^{2} - 192 T^{3} + 1446 T^{4} - 192 p T^{5} + 67 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + T - 103 T^{2} - 14 T^{3} + 7276 T^{4} - 14 p T^{5} - 103 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 43 T^{2} - 252 T^{3} - 2304 T^{4} - 252 p T^{5} + 43 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 193 T^{2} - 1530 T^{3} + 9972 T^{4} - 1530 p T^{5} + 193 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 15 T + 235 T^{2} + 2400 T^{3} + 25746 T^{4} + 2400 p T^{5} + 235 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 7 T - 107 T^{2} + 14 T^{3} + 14224 T^{4} + 14 p T^{5} - 107 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 28656 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.99305180432621922031995664618, −6.73641738802970493079039138657, −6.64624375611533058973906699704, −6.30207563323264042742466128338, −6.22758291827382797566493141903, −5.97920303257607377786024711994, −5.78132037942839086168606564902, −5.38034983685016286330471951022, −5.37218873161892973176634115012, −5.33009048859066419611178980335, −5.26111938390530634342607480321, −4.82470285454155366950724841457, −4.52643281564899588297211304365, −4.24501330327971397060252489428, −3.85539497872740389942342621209, −3.65866741619006435313047702543, −3.27161885659985407329080877494, −3.15461251450339561803003732905, −2.75459538746677241304774799013, −2.23170808545328905696212966653, −1.78626766896033821206682651831, −1.54754697755121630021168138326, −1.18072024564609385069524372375, −0.62792977295507278314955165138, −0.49432236018666601283806348925,
0.49432236018666601283806348925, 0.62792977295507278314955165138, 1.18072024564609385069524372375, 1.54754697755121630021168138326, 1.78626766896033821206682651831, 2.23170808545328905696212966653, 2.75459538746677241304774799013, 3.15461251450339561803003732905, 3.27161885659985407329080877494, 3.65866741619006435313047702543, 3.85539497872740389942342621209, 4.24501330327971397060252489428, 4.52643281564899588297211304365, 4.82470285454155366950724841457, 5.26111938390530634342607480321, 5.33009048859066419611178980335, 5.37218873161892973176634115012, 5.38034983685016286330471951022, 5.78132037942839086168606564902, 5.97920303257607377786024711994, 6.22758291827382797566493141903, 6.30207563323264042742466128338, 6.64624375611533058973906699704, 6.73641738802970493079039138657, 6.99305180432621922031995664618
