| L(s) = 1 | + 8·2-s + 48·4-s − 64·5-s + 256·8-s + 80·9-s − 512·10-s + 1.28e3·16-s + 640·18-s − 3.07e3·20-s + 1.82e3·25-s + 6.14e3·32-s + 3.84e3·36-s − 2.56e3·37-s − 1.63e4·40-s + 3.36e3·41-s − 5.12e3·45-s + 4.72e3·49-s + 1.45e4·50-s − 1.16e4·61-s + 2.86e4·64-s + 2.04e4·72-s + 1.32e4·73-s − 2.04e4·74-s − 8.19e4·80-s − 161·81-s + 2.68e4·82-s − 4.09e4·90-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s − 2.55·5-s + 4·8-s + 0.987·9-s − 5.11·10-s + 5·16-s + 1.97·18-s − 7.67·20-s + 2.91·25-s + 6·32-s + 2.96·36-s − 1.86·37-s − 10.2·40-s + 2·41-s − 2.52·45-s + 1.96·49-s + 5.83·50-s − 3.14·61-s + 7·64-s + 3.95·72-s + 2.49·73-s − 3.73·74-s − 12.7·80-s − 0.0245·81-s + 4·82-s − 5.05·90-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(6.920966999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.920966999\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 41 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 80 T^{2} + p^{8} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 32 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 4720 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 29200 T^{2} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 251120 T^{2} + p^{8} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1280 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7633840 T^{2} + p^{8} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5842 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35111600 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 17131120 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6640 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 56238800 T^{2} + p^{8} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
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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
−12.37246261797302127182394352203, −12.24169904516182496100004072612, −11.55033321927412137376891101942, −11.25494493455494294451999636668, −10.62362189259177916319024205342, −10.45898290376744742394504274869, −9.411707143803222320460994880040, −8.494198150606186588502890873951, −7.81752058549496392427116247946, −7.52521598132511700179770055482, −7.18302823894475462403837215971, −6.59968972828558604157668503324, −5.81731355471793712886017800596, −5.05758901719564710969989509734, −4.25718651448151447514291618611, −4.24671367848735121540698518826, −3.56289425308964542924419469381, −3.01035582014138925288903119558, −1.89507619058113595810280667160, −0.73757303599972518669136658957,
0.73757303599972518669136658957, 1.89507619058113595810280667160, 3.01035582014138925288903119558, 3.56289425308964542924419469381, 4.24671367848735121540698518826, 4.25718651448151447514291618611, 5.05758901719564710969989509734, 5.81731355471793712886017800596, 6.59968972828558604157668503324, 7.18302823894475462403837215971, 7.52521598132511700179770055482, 7.81752058549496392427116247946, 8.494198150606186588502890873951, 9.411707143803222320460994880040, 10.45898290376744742394504274869, 10.62362189259177916319024205342, 11.25494493455494294451999636668, 11.55033321927412137376891101942, 12.24169904516182496100004072612, 12.37246261797302127182394352203
