| L(s) = 1 | − 2·7-s + 6·9-s + 4·17-s − 25-s − 8·31-s + 4·41-s + 3·49-s − 12·63-s + 16·71-s + 20·73-s + 16·79-s + 27·81-s + 12·89-s + 4·97-s − 8·103-s − 4·113-s − 8·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2·9-s + 0.970·17-s − 1/5·25-s − 1.43·31-s + 0.624·41-s + 3/7·49-s − 1.51·63-s + 1.89·71-s + 2.34·73-s + 1.80·79-s + 3·81-s + 1.27·89-s + 0.406·97-s − 0.788·103-s − 0.376·113-s − 0.733·119-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.903103541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.903103541\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.397709449764440221055686101022, −9.081277249170364609466657343396, −8.255738821406067026695761339602, −8.081889991693258912169349168030, −7.63278042005552535079640817554, −7.28774289953926539584235465854, −6.80948385358197838069408110964, −6.75188847030154227609461940845, −6.07803381650021556992818869119, −5.79596310652948991419745354698, −5.04979541485169780751950772946, −5.02394188561978226573516139176, −4.28874508488671362090151674476, −3.88244353425025010823194155336, −3.52491755460838951078391393440, −3.21449516595576385481954250949, −2.15352078325479991227337807495, −2.08795100772617178767537612804, −1.18156123718087293315766263460, −0.66848935185036218028848394397,
0.66848935185036218028848394397, 1.18156123718087293315766263460, 2.08795100772617178767537612804, 2.15352078325479991227337807495, 3.21449516595576385481954250949, 3.52491755460838951078391393440, 3.88244353425025010823194155336, 4.28874508488671362090151674476, 5.02394188561978226573516139176, 5.04979541485169780751950772946, 5.79596310652948991419745354698, 6.07803381650021556992818869119, 6.75188847030154227609461940845, 6.80948385358197838069408110964, 7.28774289953926539584235465854, 7.63278042005552535079640817554, 8.081889991693258912169349168030, 8.255738821406067026695761339602, 9.081277249170364609466657343396, 9.397709449764440221055686101022
