L(s) = 1 | − 4-s − 9-s − 2·11-s + 16-s + 8·19-s − 12·29-s + 16·31-s + 36-s + 12·41-s + 2·44-s + 10·49-s + 16·61-s − 64-s + 12·71-s − 8·76-s − 28·79-s + 81-s + 12·89-s + 2·99-s + 12·101-s + 8·109-s + 12·116-s + 3·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1/3·9-s − 0.603·11-s + 1/4·16-s + 1.83·19-s − 2.22·29-s + 2.87·31-s + 1/6·36-s + 1.87·41-s + 0.301·44-s + 10/7·49-s + 2.04·61-s − 1/8·64-s + 1.42·71-s − 0.917·76-s − 3.15·79-s + 1/9·81-s + 1.27·89-s + 0.201·99-s + 1.19·101-s + 0.766·109-s + 1.11·116-s + 3/11·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.955676276\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.955676276\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.815626142798945392344320931911, −9.161493250100856088232914073754, −8.697682676434650456013254565807, −8.539656211641887564007120877719, −7.898714789356682017862785155492, −7.50472410226835804207180047752, −7.47056351506611648043844488695, −6.81727640020239548589089864255, −6.25910130489876625955844609685, −5.72604065846186135363486619021, −5.62626800481337350779750276752, −5.00495502119669651819120876736, −4.74478526447514835009412810378, −3.94382568015696916455588784502, −3.83998232499477175671902486222, −3.02920176134488095376951961213, −2.68733075069823990382751008584, −2.13343775033899613311035906501, −1.14299162909676704652205455168, −0.62995539807950837900953515777,
0.62995539807950837900953515777, 1.14299162909676704652205455168, 2.13343775033899613311035906501, 2.68733075069823990382751008584, 3.02920176134488095376951961213, 3.83998232499477175671902486222, 3.94382568015696916455588784502, 4.74478526447514835009412810378, 5.00495502119669651819120876736, 5.62626800481337350779750276752, 5.72604065846186135363486619021, 6.25910130489876625955844609685, 6.81727640020239548589089864255, 7.47056351506611648043844488695, 7.50472410226835804207180047752, 7.898714789356682017862785155492, 8.539656211641887564007120877719, 8.697682676434650456013254565807, 9.161493250100856088232914073754, 9.815626142798945392344320931911