| L(s) = 1 | − 3-s − 3·5-s − 9-s − 2·11-s − 4·13-s + 3·15-s + 6·17-s − 6·19-s − 3·23-s + 25-s − 6·29-s − 15·31-s + 2·33-s − 37-s + 4·39-s − 6·43-s + 3·45-s − 8·47-s − 14·49-s − 6·51-s + 6·55-s + 6·57-s + 13·59-s + 6·61-s + 12·65-s − 3·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.774·15-s + 1.45·17-s − 1.37·19-s − 0.625·23-s + 1/5·25-s − 1.11·29-s − 2.69·31-s + 0.348·33-s − 0.164·37-s + 0.640·39-s − 0.914·43-s + 0.447·45-s − 1.16·47-s − 2·49-s − 0.840·51-s + 0.809·55-s + 0.794·57-s + 1.69·59-s + 0.768·61-s + 1.48·65-s − 0.366·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 15 T + 114 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 13 T + 122 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 19 T + 194 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 176 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−9.994066662245861355755360111118, −9.965088507638647073968732990367, −9.623515760419739714697837517073, −8.656929244624234630824060750928, −8.530945589208049100672885995963, −8.011978207178670625658766419097, −7.49674853250841138701759634638, −7.35414587092176273137964814482, −6.84666074039592107350230004766, −6.09650404237675435814866983305, −5.65703739115574662441780954724, −5.28888958868363812420471932101, −4.78554179254037215939622947776, −4.17676354055744813843210672677, −3.48678432040528483062222868330, −3.41502641256963178080103047658, −2.30906917918481945396602919838, −1.72303323645901048412833725201, 0, 0,
1.72303323645901048412833725201, 2.30906917918481945396602919838, 3.41502641256963178080103047658, 3.48678432040528483062222868330, 4.17676354055744813843210672677, 4.78554179254037215939622947776, 5.28888958868363812420471932101, 5.65703739115574662441780954724, 6.09650404237675435814866983305, 6.84666074039592107350230004766, 7.35414587092176273137964814482, 7.49674853250841138701759634638, 8.011978207178670625658766419097, 8.530945589208049100672885995963, 8.656929244624234630824060750928, 9.623515760419739714697837517073, 9.965088507638647073968732990367, 9.994066662245861355755360111118
