| L(s) = 1 | − 3·3-s − 3·5-s + 2·7-s + 4·9-s + 2·11-s − 6·13-s + 9·15-s − 2·17-s − 12·19-s − 6·21-s + 2·23-s − 6·27-s − 10·29-s − 17·31-s − 6·33-s − 6·35-s + 18·39-s + 41-s − 5·43-s − 12·45-s + 20·47-s + 3·49-s + 6·51-s + 5·53-s − 6·55-s + 36·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 0.755·7-s + 4/3·9-s + 0.603·11-s − 1.66·13-s + 2.32·15-s − 0.485·17-s − 2.75·19-s − 1.30·21-s + 0.417·23-s − 1.15·27-s − 1.85·29-s − 3.05·31-s − 1.04·33-s − 1.01·35-s + 2.88·39-s + 0.156·41-s − 0.762·43-s − 1.78·45-s + 2.91·47-s + 3/7·49-s + 0.840·51-s + 0.686·53-s − 0.809·55-s + 4.76·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 17 T + 131 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_4$ | \( 1 + 5 T + 89 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 109 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 125 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 151 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 85 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 221 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−10.85696882384837336755774984979, −10.79420332340129358871089145272, −10.08964048887404750722789904321, −9.440023554094045298409228756394, −8.838366900915876296429655943828, −8.739793279493932851068109336666, −7.76481959306128598922806051066, −7.50073954977690814783436592085, −7.15387354484516515755702646157, −6.70385753197726502977657401306, −5.85128632622433463351478609513, −5.69114875933762450620973709082, −5.09277283063707884389548729149, −4.47554244256704604327094464595, −4.01621350754607537551052472636, −3.80293690894226375210320066440, −2.32822010793954904243592174695, −1.80826892973584175146586308601, 0, 0,
1.80826892973584175146586308601, 2.32822010793954904243592174695, 3.80293690894226375210320066440, 4.01621350754607537551052472636, 4.47554244256704604327094464595, 5.09277283063707884389548729149, 5.69114875933762450620973709082, 5.85128632622433463351478609513, 6.70385753197726502977657401306, 7.15387354484516515755702646157, 7.50073954977690814783436592085, 7.76481959306128598922806051066, 8.739793279493932851068109336666, 8.838366900915876296429655943828, 9.440023554094045298409228756394, 10.08964048887404750722789904321, 10.79420332340129358871089145272, 10.85696882384837336755774984979
