| L(s) = 1 | + 4-s + 4·7-s − 3·16-s + 6·17-s + 8·19-s + 12·23-s − 2·25-s + 4·28-s − 6·29-s + 4·31-s + 7·49-s − 12·53-s − 7·64-s + 8·67-s + 6·68-s + 8·76-s − 24·83-s + 6·89-s + 12·92-s + 4·97-s − 2·100-s + 20·103-s + 10·109-s − 12·112-s − 6·116-s + 24·119-s − 13·121-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s − 3/4·16-s + 1.45·17-s + 1.83·19-s + 2.50·23-s − 2/5·25-s + 0.755·28-s − 1.11·29-s + 0.718·31-s + 49-s − 1.64·53-s − 7/8·64-s + 0.977·67-s + 0.727·68-s + 0.917·76-s − 2.63·83-s + 0.635·89-s + 1.25·92-s + 0.406·97-s − 1/5·100-s + 1.97·103-s + 0.957·109-s − 1.13·112-s − 0.557·116-s + 2.20·119-s − 1.18·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.312132288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.312132288\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 31 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.213991263683801851847537832529, −7.74192889391156366478542237048, −7.42873920554066937283486881965, −7.22789184646972448097877280741, −6.59288204234028210857954290808, −6.01332242961889888110924651535, −5.40288069193282690804591691562, −5.12677391812262806360162256580, −4.79498391916885294645619856154, −4.13987747017055240637941010851, −3.25628208292663380469786100187, −3.10095685377418688712316967565, −2.25481857866205328827216224936, −1.46284674926049666159918523925, −1.04329085035080453884056849465,
1.04329085035080453884056849465, 1.46284674926049666159918523925, 2.25481857866205328827216224936, 3.10095685377418688712316967565, 3.25628208292663380469786100187, 4.13987747017055240637941010851, 4.79498391916885294645619856154, 5.12677391812262806360162256580, 5.40288069193282690804591691562, 6.01332242961889888110924651535, 6.59288204234028210857954290808, 7.22789184646972448097877280741, 7.42873920554066937283486881965, 7.74192889391156366478542237048, 8.213991263683801851847537832529
