| L(s) = 1 | + 12·5-s − 243·9-s − 5.30e3·13-s − 1.44e4·17-s − 3.11e4·25-s + 2.31e4·29-s + 4.46e4·37-s + 2.07e5·41-s − 2.91e3·45-s + 1.94e5·49-s + 3.36e5·53-s − 5.20e5·61-s − 6.36e4·65-s − 7.91e5·73-s + 5.90e4·81-s − 1.72e5·85-s − 5.03e5·89-s + 1.03e6·97-s + 2.04e6·101-s + 1.90e6·109-s − 2.42e5·113-s + 1.28e6·117-s + 3.12e6·121-s − 5.61e5·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.0959·5-s − 1/3·9-s − 2.41·13-s − 2.93·17-s − 1.99·25-s + 0.947·29-s + 0.882·37-s + 3.00·41-s − 0.0319·45-s + 1.65·49-s + 2.26·53-s − 2.29·61-s − 0.231·65-s − 2.03·73-s + 1/9·81-s − 0.281·85-s − 0.714·89-s + 1.13·97-s + 1.98·101-s + 1.47·109-s − 0.167·113-s + 0.805·117-s + 1.76·121-s − 0.287·125-s + 0.0909·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8634100806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8634100806\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 6 T + p^{6} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 194930 T^{2} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3127970 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2654 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 7206 T + p^{6} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 81557954 T^{2} + p^{12} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8747422 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 11550 T + p^{6} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1702033490 T^{2} + p^{12} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 22346 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 103626 T + p^{6} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3585198814 T^{2} + p^{12} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4308279550 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 168462 T + p^{6} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 20657330 p^{2} T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 70 p^{2} T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 80375086466 T^{2} + p^{12} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 245662064350 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 395918 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 176211604754 T^{2} + p^{12} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 645933881666 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 251886 T + p^{6} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 517474 T + p^{6} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.10830395803282621320207669080, −13.85127807765206180937990588976, −13.81306630991808129319686103627, −12.87614252940234737151589214137, −12.45493350383591380837356290411, −11.62780400857951881692957118588, −11.38137728038245121524701519841, −10.42413192457682826110629185955, −9.974065668017492652069422815849, −9.036925433463367198350428249143, −8.923577798764588973130695718964, −7.53416741076928938452062661581, −7.45021523197529570861112552693, −6.37188631452682379179796338036, −5.72358008288561230392697284984, −4.53683903949108683116077454354, −4.30485432199147628607100982584, −2.44613130536843901296785577287, −2.35199887468584721278530252211, −0.39503320868233161377919572993,
0.39503320868233161377919572993, 2.35199887468584721278530252211, 2.44613130536843901296785577287, 4.30485432199147628607100982584, 4.53683903949108683116077454354, 5.72358008288561230392697284984, 6.37188631452682379179796338036, 7.45021523197529570861112552693, 7.53416741076928938452062661581, 8.923577798764588973130695718964, 9.036925433463367198350428249143, 9.974065668017492652069422815849, 10.42413192457682826110629185955, 11.38137728038245121524701519841, 11.62780400857951881692957118588, 12.45493350383591380837356290411, 12.87614252940234737151589214137, 13.81306630991808129319686103627, 13.85127807765206180937990588976, 15.10830395803282621320207669080
