L(s) = 1 | − 4-s − 9-s + 8·11-s + 16-s − 8·19-s − 12·29-s − 16·31-s + 36-s − 12·41-s − 8·44-s + 14·49-s − 8·59-s − 4·61-s − 64-s + 32·71-s + 8·76-s + 32·79-s + 81-s − 20·89-s − 8·99-s − 4·101-s + 20·109-s + 12·116-s + 26·121-s + 16·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1/3·9-s + 2.41·11-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 − 1.20·44-s + 2·49-s − 1.04·59-s − 0.512·61-s − 1/8·64-s + 3.79·71-s + 0.917·76-s + 3.60·79-s + 1/9·81-s − 2.11·89-s − 0.804·99-s − 0.398·101-s + 1.91·109-s + 1.11·116-s + 2.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.254680078\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.254680078\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−9.236169835844586346852186900313, −9.154252155014664858822159767044, −8.589440121104046330999730805296, −8.487864019511558050260952450192, −7.79524108488206212661689876148, −7.43308946362867899085930968108, −6.82520505257646537146941498508, −6.79355452885207774902165474077, −6.12404854671603562682731776745, −5.94652991610837050895795060529, −5.21510627854161874037652402094, −5.12496116293216192488918629710, −4.26387253987100039449342818826, −3.95441788940311451170544654144, −3.55723636751264796696742918548, −3.50815152837742043223745463572, −2.21065775232864566041423645635, −1.98132603108891222983698033928, −1.41712201181598111067079437900, −0.41220225760447435717141390099,
0.41220225760447435717141390099, 1.41712201181598111067079437900, 1.98132603108891222983698033928, 2.21065775232864566041423645635, 3.50815152837742043223745463572, 3.55723636751264796696742918548, 3.95441788940311451170544654144, 4.26387253987100039449342818826, 5.12496116293216192488918629710, 5.21510627854161874037652402094, 5.94652991610837050895795060529, 6.12404854671603562682731776745, 6.79355452885207774902165474077, 6.82520505257646537146941498508, 7.43308946362867899085930968108, 7.79524108488206212661689876148, 8.487864019511558050260952450192, 8.589440121104046330999730805296, 9.154252155014664858822159767044, 9.236169835844586346852186900313