| L(s) = 1 | − 3·5-s − 2·7-s + 3·11-s + 4·13-s + 3·17-s − 10·19-s − 9·23-s + 5·25-s + 6·29-s − 4·31-s + 6·35-s + 4·37-s − 15·41-s − 43-s + 3·49-s + 6·53-s − 9·55-s − 3·59-s − 11·61-s − 12·65-s − 13·67-s + 3·71-s + 7·73-s − 6·77-s − 7·79-s + 12·83-s − 9·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s + 0.904·11-s + 1.10·13-s + 0.727·17-s − 2.29·19-s − 1.87·23-s + 25-s + 1.11·29-s − 0.718·31-s + 1.01·35-s + 0.657·37-s − 2.34·41-s − 0.152·43-s + 3/7·49-s + 0.824·53-s − 1.21·55-s − 0.390·59-s − 1.40·61-s − 1.48·65-s − 1.58·67-s + 0.356·71-s + 0.819·73-s − 0.683·77-s − 0.787·79-s + 1.31·83-s − 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82301184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82301184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 15 T + 130 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 96 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 120 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−7.49819129050945152678159805660, −7.33904782818921849819659807796, −6.74077671050710717338325794918, −6.49012786480387222411372933903, −6.24851272659363846945735161537, −6.11430880883318722000610078856, −5.54950983348875706202560869292, −5.14095031024772881740613222428, −4.47326216120974344022097209619, −4.38272672290212644294860081696, −3.93425098518615902304588292574, −3.80099467177175591140117293803, −3.20087802776295477107327487739, −3.19364928163553671669682340061, −2.38679141024468417545491990507, −1.94077573619259854784712224247, −1.47798059349005779261588754731, −0.912734859987117860545957080215, 0, 0,
0.912734859987117860545957080215, 1.47798059349005779261588754731, 1.94077573619259854784712224247, 2.38679141024468417545491990507, 3.19364928163553671669682340061, 3.20087802776295477107327487739, 3.80099467177175591140117293803, 3.93425098518615902304588292574, 4.38272672290212644294860081696, 4.47326216120974344022097209619, 5.14095031024772881740613222428, 5.54950983348875706202560869292, 6.11430880883318722000610078856, 6.24851272659363846945735161537, 6.49012786480387222411372933903, 6.74077671050710717338325794918, 7.33904782818921849819659807796, 7.49819129050945152678159805660
