| L(s) = 1 | − 48·13-s − 48·17-s + 20·29-s − 48·37-s + 68·41-s + 50·49-s + 96·53-s + 140·61-s − 96·73-s − 28·89-s − 192·97-s + 76·101-s − 332·109-s + 336·113-s + 50·121-s + ⋯ |
| L(s) = 1 | − 3.69·13-s − 2.82·17-s + 0.689·29-s − 1.29·37-s + 1.65·41-s + 1.02·49-s + 1.81·53-s + 2.29·61-s − 1.31·73-s − 0.314·89-s − 1.97·97-s + 0.752·101-s − 3.04·109-s + 2.97·113-s + 0.413·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8689083810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8689083810\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 26 T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 142 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1730 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4370 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6770 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 866 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 7010 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10754 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 5666 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 96 T + p^{2} T^{2} )^{2} \) |
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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
−8.552985955710740973447221526840, −8.304145573618465820567822761120, −7.57544076945079864059198651257, −7.53250730343261665393086647451, −6.86947418231085623223033476707, −6.79338744745835797635317688067, −6.78404217668536048186707309371, −5.78518679848958454011645447753, −5.47176926863284911386384838307, −5.23428035108756472616833304679, −4.51156232934070883087477692047, −4.46041523666332039723405815156, −4.26351061155675996377485518456, −3.48582594539429197265039817571, −2.66463999562189412348199221781, −2.57673871663631231776383350311, −2.22875678674716393255746322758, −1.83256571398085651358688912187, −0.71577996072752028088685518931, −0.26637459699049133777464369688,
0.26637459699049133777464369688, 0.71577996072752028088685518931, 1.83256571398085651358688912187, 2.22875678674716393255746322758, 2.57673871663631231776383350311, 2.66463999562189412348199221781, 3.48582594539429197265039817571, 4.26351061155675996377485518456, 4.46041523666332039723405815156, 4.51156232934070883087477692047, 5.23428035108756472616833304679, 5.47176926863284911386384838307, 5.78518679848958454011645447753, 6.78404217668536048186707309371, 6.79338744745835797635317688067, 6.86947418231085623223033476707, 7.53250730343261665393086647451, 7.57544076945079864059198651257, 8.304145573618465820567822761120, 8.552985955710740973447221526840
