| L(s) = 1 | + 3-s − 4·5-s + 5·7-s − 6·11-s − 10·13-s − 4·15-s − 2·17-s − 19-s + 5·21-s − 6·23-s + 5·25-s − 27-s − 3·31-s − 6·33-s − 20·35-s + 3·37-s − 10·39-s − 12·41-s + 10·43-s − 4·47-s + 18·49-s − 2·51-s − 6·53-s + 24·55-s − 57-s + 6·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1.88·7-s − 1.80·11-s − 2.77·13-s − 1.03·15-s − 0.485·17-s − 0.229·19-s + 1.09·21-s − 1.25·23-s + 25-s − 0.192·27-s − 0.538·31-s − 1.04·33-s − 3.38·35-s + 0.493·37-s − 1.60·39-s − 1.87·41-s + 1.52·43-s − 0.583·47-s + 18/7·49-s − 0.280·51-s − 0.824·53-s + 3.23·55-s − 0.132·57-s + 0.781·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.419382354141753284742995101688, −8.888894369879400097993493848140, −8.223373703662598902568952040242, −8.137627364123077207530094041446, −7.73981743416146513036747746327, −7.73430200867319171148577948312, −7.18554149881087138515494660794, −6.94175021694050638782766532680, −5.96573570114839498057923853526, −5.39448727056468430255608451844, −5.01923809027054233052667864290, −4.74595426399781404789886099900, −4.27341296974478721417591623125, −3.97631658247478653323370928207, −3.16042618390013366449468201205, −2.49305908448567367030785356109, −2.35840850751405595125872558675, −1.61916457107724997243687807398, 0, 0,
1.61916457107724997243687807398, 2.35840850751405595125872558675, 2.49305908448567367030785356109, 3.16042618390013366449468201205, 3.97631658247478653323370928207, 4.27341296974478721417591623125, 4.74595426399781404789886099900, 5.01923809027054233052667864290, 5.39448727056468430255608451844, 5.96573570114839498057923853526, 6.94175021694050638782766532680, 7.18554149881087138515494660794, 7.73430200867319171148577948312, 7.73981743416146513036747746327, 8.137627364123077207530094041446, 8.223373703662598902568952040242, 8.888894369879400097993493848140, 9.419382354141753284742995101688
