| L(s) = 1 | − 3·3-s − 5-s − 7-s + 6·9-s + 4·13-s + 3·15-s − 12·17-s − 4·19-s + 3·21-s − 3·23-s − 9·27-s − 3·29-s − 10·31-s + 35-s − 20·37-s − 12·39-s − 9·41-s − 4·43-s − 6·45-s + 9·47-s + 7·49-s + 36·51-s − 12·53-s + 12·57-s − 6·59-s + 61-s − 6·63-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s + 1.10·13-s + 0.774·15-s − 2.91·17-s − 0.917·19-s + 0.654·21-s − 0.625·23-s − 1.73·27-s − 0.557·29-s − 1.79·31-s + 0.169·35-s − 3.28·37-s − 1.92·39-s − 1.40·41-s − 0.609·43-s − 0.894·45-s + 1.31·47-s + 49-s + 5.04·51-s − 1.64·53-s + 1.58·57-s − 0.781·59-s + 0.128·61-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.41940890496104677867537904419, −10.12004617562558317612234555629, −9.049173408442582028060569306632, −9.012758654883809749389358977690, −8.693431316048271361538352350262, −8.050711853716119625023850986512, −7.15346157054947000600534800765, −7.13070494244411027727927907447, −6.47477624670373675122207936470, −6.41203781582680632548827353130, −5.55427569378275538933118585294, −5.51434012765184138238758748848, −4.49613787151169548431890054640, −4.47202894275194399058100555398, −3.77857782832053877074108713568, −3.28228619378010980214582817528, −1.94778818271643003958566757316, −1.73870141948584193105546101954, 0, 0,
1.73870141948584193105546101954, 1.94778818271643003958566757316, 3.28228619378010980214582817528, 3.77857782832053877074108713568, 4.47202894275194399058100555398, 4.49613787151169548431890054640, 5.51434012765184138238758748848, 5.55427569378275538933118585294, 6.41203781582680632548827353130, 6.47477624670373675122207936470, 7.13070494244411027727927907447, 7.15346157054947000600534800765, 8.050711853716119625023850986512, 8.693431316048271361538352350262, 9.012758654883809749389358977690, 9.049173408442582028060569306632, 10.12004617562558317612234555629, 10.41940890496104677867537904419
