| L(s) = 1 | + 2-s − 2·3-s − 2·4-s + 2·5-s − 2·6-s − 3·8-s − 3·9-s + 2·10-s + 2·11-s + 4·12-s − 4·15-s + 16-s − 4·17-s − 3·18-s + 6·19-s − 4·20-s + 2·22-s − 4·23-s + 6·24-s − 7·25-s + 14·27-s − 4·30-s − 4·31-s + 2·32-s − 4·33-s − 4·34-s + 6·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 4-s + 0.894·5-s − 0.816·6-s − 1.06·8-s − 9-s + 0.632·10-s + 0.603·11-s + 1.15·12-s − 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.707·18-s + 1.37·19-s − 0.894·20-s + 0.426·22-s − 0.834·23-s + 1.22·24-s − 7/5·25-s + 2.69·27-s − 0.730·30-s − 0.718·31-s + 0.353·32-s − 0.696·33-s − 0.685·34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 27 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 165 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 77 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 127 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 175 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 317 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.143896342821664412650679212473, −8.742788632769436915187458469362, −8.093638656745808009727001966605, −7.88642658358355077153591660162, −7.42690811120597763083642518078, −6.57278455772427309319060051389, −6.31133638529926989861796396356, −6.18370765298369500248541280244, −5.66372663084856165935161771579, −5.27791679221505233317792505211, −5.10554524586296652203508336762, −4.67253271559183711381906114086, −3.97841113105928143708392076201, −3.83097095675924780696787431610, −3.02754831591257643973819566004, −2.71729396873265872387249726738, −1.84903654838820738793935898387, −1.28444496299652387573861537597, 0, 0,
1.28444496299652387573861537597, 1.84903654838820738793935898387, 2.71729396873265872387249726738, 3.02754831591257643973819566004, 3.83097095675924780696787431610, 3.97841113105928143708392076201, 4.67253271559183711381906114086, 5.10554524586296652203508336762, 5.27791679221505233317792505211, 5.66372663084856165935161771579, 6.18370765298369500248541280244, 6.31133638529926989861796396356, 6.57278455772427309319060051389, 7.42690811120597763083642518078, 7.88642658358355077153591660162, 8.093638656745808009727001966605, 8.742788632769436915187458469362, 9.143896342821664412650679212473
