| L(s) = 1 | + 18·5-s − 10·7-s − 4·11-s + 26·13-s + 52·17-s − 142·19-s + 280·23-s + 10·25-s + 424·29-s − 178·31-s − 180·35-s + 136·37-s + 226·41-s − 72·43-s − 176·47-s − 186·49-s + 788·53-s − 72·55-s − 728·59-s − 736·61-s + 468·65-s + 1.05e3·67-s + 660·71-s + 24·73-s + 40·77-s + 40·79-s − 1.34e3·83-s + ⋯ |
| L(s) = 1 | + 1.60·5-s − 0.539·7-s − 0.109·11-s + 0.554·13-s + 0.741·17-s − 1.71·19-s + 2.53·23-s + 2/25·25-s + 2.71·29-s − 1.03·31-s − 0.869·35-s + 0.604·37-s + 0.860·41-s − 0.255·43-s − 0.546·47-s − 0.542·49-s + 2.04·53-s − 0.176·55-s − 1.60·59-s − 1.54·61-s + 0.893·65-s + 1.92·67-s + 1.10·71-s + 0.0384·73-s + 0.0592·77-s + 0.0569·79-s − 1.77·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.716507757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.716507757\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 18 T + 314 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T + 286 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T - 666 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 52 T + 8054 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 142 T + 16702 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 280 T + 41486 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 424 T + 93110 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 178 T + 48990 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 136 T + 56358 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 226 T + 144474 T^{2} - 226 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 72 T + 28662 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 176 T - 29410 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 788 T + 451902 T^{2} - 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 728 T + 542166 T^{2} + 728 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 736 T + 564838 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1054 T + 684622 T^{2} - 1054 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 660 T + 721294 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T - 59650 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 40 T - 308514 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1344 T + 1516550 T^{2} + 1344 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 314 T + 619250 T^{2} + 314 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 264 T + 1841070 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.761056341602527945675650061066, −9.600076179397394488234730721532, −8.962232851793488429773739739044, −8.895911072595924749869130179301, −8.220984110160874276207255023313, −7.954194627891241501068607939755, −7.13767670878649367969540790457, −6.78834215771319457470197041577, −6.43181432700920144276756550578, −6.05286430945293843138644501046, −5.59240814163298022952269914140, −5.24698985792822060531888437767, −4.51232406615639476375093926393, −4.28304395088599863683676105412, −3.20842440946067003174356554093, −3.10694584939230655518767094456, −2.34400674223731783986628831367, −1.88325420162029268508603276097, −1.13746870841726514792562803097, −0.60273264324237862360217974959,
0.60273264324237862360217974959, 1.13746870841726514792562803097, 1.88325420162029268508603276097, 2.34400674223731783986628831367, 3.10694584939230655518767094456, 3.20842440946067003174356554093, 4.28304395088599863683676105412, 4.51232406615639476375093926393, 5.24698985792822060531888437767, 5.59240814163298022952269914140, 6.05286430945293843138644501046, 6.43181432700920144276756550578, 6.78834215771319457470197041577, 7.13767670878649367969540790457, 7.954194627891241501068607939755, 8.220984110160874276207255023313, 8.895911072595924749869130179301, 8.962232851793488429773739739044, 9.600076179397394488234730721532, 9.761056341602527945675650061066
