| L(s) = 1 | + 162·3-s + 596·5-s − 5.30e3·7-s + 1.96e4·9-s − 2.40e4·11-s − 9.06e3·13-s + 9.65e4·15-s − 4.36e5·17-s − 2.48e5·19-s − 8.59e5·21-s − 3.77e5·23-s − 3.53e6·25-s + 2.12e6·27-s − 1.56e5·29-s − 4.77e6·31-s − 3.89e6·33-s − 3.16e6·35-s + 7.54e6·37-s − 1.46e6·39-s + 1.41e6·41-s − 4.43e7·43-s + 1.17e7·45-s − 5.67e7·47-s − 2.89e7·49-s − 7.07e7·51-s + 1.03e8·53-s − 1.43e7·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.426·5-s − 0.834·7-s + 9-s − 0.494·11-s − 0.0879·13-s + 0.492·15-s − 1.26·17-s − 0.437·19-s − 0.964·21-s − 0.281·23-s − 1.80·25-s + 0.769·27-s − 0.0411·29-s − 0.928·31-s − 0.571·33-s − 0.356·35-s + 0.662·37-s − 0.101·39-s + 0.0781·41-s − 1.97·43-s + 0.426·45-s − 1.69·47-s − 0.718·49-s − 1.46·51-s + 1.79·53-s − 0.210·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 596 T + 777814 p T^{2} - 596 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5304 T + 8158706 p T^{2} + 5304 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2184 p T + 1939688422 T^{2} + 2184 p^{10} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 9060 T - 4634391778 T^{2} + 9060 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 436460 T + 174966323078 T^{2} + 436460 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 248424 T + 529278296438 T^{2} + 248424 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 377184 T + 2076702085294 T^{2} + 377184 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 156812 T + 23408085454238 T^{2} + 156812 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4771944 T + 31423242616526 T^{2} + 4771944 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7547324 T + 6692224350966 p T^{2} - 7547324 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1414628 T + 504550239988118 T^{2} - 1414628 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 44331288 T + 1460673636952358 T^{2} + 44331288 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 56719632 T + 2828246698281886 T^{2} + 56719632 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 103087684 T + 9177010617654446 T^{2} - 103087684 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 199358472 T + 23347455557155270 T^{2} + 199358472 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 83781004 T + 24811404918762462 T^{2} - 83781004 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 352749048 T + 84868407427389014 T^{2} + 352749048 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 470321664 T + 128531200755673870 T^{2} + 470321664 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 232248972 T + 91294387533143606 T^{2} + 232248972 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 543584808 T + 243080477878069550 T^{2} + 543584808 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11108760 p T + 583333230061860982 T^{2} + 11108760 p^{10} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 404785748 T + 713899124440367894 T^{2} - 404785748 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1436252580 T + 1127643970262491910 T^{2} - 1436252580 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−11.69637683913323632435808748547, −11.59222511353643826488443437221, −10.46370961503633171923414874726, −10.23239116229748713791905687730, −9.622259728780904909346614182871, −9.191802121087895001005372415443, −8.615432268363724497469388193319, −8.100105780597879544119167368911, −7.42531966825134483892486181276, −6.89578036115445115760544526918, −6.16691101099168966451199148052, −5.69255987545532271144509043040, −4.59451563236916303697173024185, −4.16723525126353681724692965199, −3.23663598557852347595122602949, −2.86419018716876155754553305985, −1.88018621579946704307463455177, −1.69520951224430454505075090568, 0, 0,
1.69520951224430454505075090568, 1.88018621579946704307463455177, 2.86419018716876155754553305985, 3.23663598557852347595122602949, 4.16723525126353681724692965199, 4.59451563236916303697173024185, 5.69255987545532271144509043040, 6.16691101099168966451199148052, 6.89578036115445115760544526918, 7.42531966825134483892486181276, 8.100105780597879544119167368911, 8.615432268363724497469388193319, 9.191802121087895001005372415443, 9.622259728780904909346614182871, 10.23239116229748713791905687730, 10.46370961503633171923414874726, 11.59222511353643826488443437221, 11.69637683913323632435808748547
