| L(s) = 1 | − 2·5-s − 8·13-s + 4·17-s + 3·25-s + 12·29-s − 16·37-s + 16·41-s + 6·49-s + 12·53-s − 20·61-s + 16·65-s + 12·73-s − 8·85-s + 8·89-s + 4·97-s − 20·101-s + 12·109-s + 28·113-s − 2·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2.21·13-s + 0.970·17-s + 3/5·25-s + 2.22·29-s − 2.63·37-s + 2.49·41-s + 6/7·49-s + 1.64·53-s − 2.56·61-s + 1.98·65-s + 1.40·73-s − 0.867·85-s + 0.847·89-s + 0.406·97-s − 1.99·101-s + 1.14·109-s + 2.63·113-s − 0.181·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.495965083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.495965083\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.800465784279761253233886456301, −8.763026357627653920761174594592, −7.983123196894747933221895244336, −7.86763016887871456690762651243, −7.38753611913052587246135534692, −7.29049715075276746344719229442, −6.75173052657709329743568618103, −6.45068652784930501555226337243, −5.82443657124409803481935673694, −5.43984296848743151557268911494, −4.88235030527088699767275715742, −4.86617337784701656310434321388, −4.17430382809723215490575683198, −3.96392297175055580942598880666, −3.15695099480266029323052790721, −2.97838812575968238344653964471, −2.43759437446010947489514323098, −1.92082155193179561462698588211, −1.01160359107600223613547382503, −0.46142751098661565358788122608,
0.46142751098661565358788122608, 1.01160359107600223613547382503, 1.92082155193179561462698588211, 2.43759437446010947489514323098, 2.97838812575968238344653964471, 3.15695099480266029323052790721, 3.96392297175055580942598880666, 4.17430382809723215490575683198, 4.86617337784701656310434321388, 4.88235030527088699767275715742, 5.43984296848743151557268911494, 5.82443657124409803481935673694, 6.45068652784930501555226337243, 6.75173052657709329743568618103, 7.29049715075276746344719229442, 7.38753611913052587246135534692, 7.86763016887871456690762651243, 7.983123196894747933221895244336, 8.763026357627653920761174594592, 8.800465784279761253233886456301
