| L(s) = 1 | − 2·5-s + 2·9-s − 6·13-s + 8·17-s − 6·25-s + 4·29-s − 8·37-s − 16·41-s − 4·45-s − 49-s − 8·53-s + 14·61-s + 12·65-s + 4·73-s − 5·81-s − 16·85-s + 4·89-s − 10·101-s + 12·109-s − 24·113-s − 12·117-s − 6·121-s + 22·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2/3·9-s − 1.66·13-s + 1.94·17-s − 6/5·25-s + 0.742·29-s − 1.31·37-s − 2.49·41-s − 0.596·45-s − 1/7·49-s − 1.09·53-s + 1.79·61-s + 1.48·65-s + 0.468·73-s − 5/9·81-s − 1.73·85-s + 0.423·89-s − 0.995·101-s + 1.14·109-s − 2.25·113-s − 1.10·117-s − 0.545·121-s + 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.748206056161590976969499910612, −8.179946801866340062035204710361, −7.923882784723258668568775098471, −7.46828797932983835870230236609, −7.00019966923049374940816314313, −6.65709825199044769441276010201, −5.81600256753682635993897266731, −5.15713562804462145781231010951, −4.98993033938331204401775177520, −4.19620760856384859287607372494, −3.60168699871114817668965295078, −3.19883944655404546512903236127, −2.26041592348211683575803705157, −1.39860853959888617576813150797, 0,
1.39860853959888617576813150797, 2.26041592348211683575803705157, 3.19883944655404546512903236127, 3.60168699871114817668965295078, 4.19620760856384859287607372494, 4.98993033938331204401775177520, 5.15713562804462145781231010951, 5.81600256753682635993897266731, 6.65709825199044769441276010201, 7.00019966923049374940816314313, 7.46828797932983835870230236609, 7.923882784723258668568775098471, 8.179946801866340062035204710361, 8.748206056161590976969499910612
