| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·11-s + 5·16-s − 8·17-s + 8·19-s − 4·22-s − 4·23-s − 10·25-s + 8·31-s + 6·32-s − 16·34-s − 4·37-s + 16·38-s − 8·41-s − 20·43-s − 6·44-s − 8·46-s − 20·50-s − 4·53-s − 8·59-s + 16·62-s + 7·64-s − 24·68-s − 16·71-s + 8·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s + 5/4·16-s − 1.94·17-s + 1.83·19-s − 0.852·22-s − 0.834·23-s − 2·25-s + 1.43·31-s + 1.06·32-s − 2.74·34-s − 0.657·37-s + 2.59·38-s − 1.24·41-s − 3.04·43-s − 0.904·44-s − 1.17·46-s − 2.82·50-s − 0.549·53-s − 1.04·59-s + 2.03·62-s + 7/8·64-s − 2.91·68-s − 1.89·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 28 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 92 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 180 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
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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
−7.31821021334566361084735667859, −7.09913476350978606604928664483, −6.71609151954282017318310140055, −6.44713373955690254987157833210, −6.02133623511271840062019651302, −5.88928797115136615471746419813, −5.21806493573890461280459696776, −5.17181292767253348761792101856, −4.66283279238377383597084199321, −4.59075600414365518391393782159, −3.84091260915414554593211408866, −3.81224155942979933118912245834, −3.16970989580926244640150791255, −3.07157399126680664505266356566, −2.33211737022148911476889012878, −2.24564842612789629388186244474, −1.46740652730559253332895514935, −1.43068313812927773263926636795, 0, 0,
1.43068313812927773263926636795, 1.46740652730559253332895514935, 2.24564842612789629388186244474, 2.33211737022148911476889012878, 3.07157399126680664505266356566, 3.16970989580926244640150791255, 3.81224155942979933118912245834, 3.84091260915414554593211408866, 4.59075600414365518391393782159, 4.66283279238377383597084199321, 5.17181292767253348761792101856, 5.21806493573890461280459696776, 5.88928797115136615471746419813, 6.02133623511271840062019651302, 6.44713373955690254987157833210, 6.71609151954282017318310140055, 7.09913476350978606604928664483, 7.31821021334566361084735667859
