| L(s) = 1 | + 2-s + 4-s − 2·5-s + 2·7-s + 8-s + 3·9-s − 2·10-s − 2·11-s + 2·14-s + 16-s + 3·18-s + 12·19-s − 2·20-s − 2·22-s − 7·25-s + 2·28-s + 32-s − 4·35-s + 3·36-s + 6·37-s + 12·38-s − 2·40-s − 10·43-s − 2·44-s − 6·45-s − 11·49-s − 7·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.755·7-s + 0.353·8-s + 9-s − 0.632·10-s − 0.603·11-s + 0.534·14-s + 1/4·16-s + 0.707·18-s + 2.75·19-s − 0.447·20-s − 0.426·22-s − 7/5·25-s + 0.377·28-s + 0.176·32-s − 0.676·35-s + 1/2·36-s + 0.986·37-s + 1.94·38-s − 0.316·40-s − 1.52·43-s − 0.301·44-s − 0.894·45-s − 1.57·49-s − 0.989·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 654368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 654368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.255340507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.255340507\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.213624087600461730406930031285, −7.70041537652024622615801427573, −7.45985560667588965978165906144, −7.23852620954424289310886833172, −6.63762568339569193603820373666, −5.88908450915949561478136129218, −5.53658965585636024903423395160, −5.04533228961993992950876423947, −4.61153453491182141362175201622, −4.17049361604183668150248255904, −3.44686426377115101001934128516, −3.32154928460028163630088476789, −2.34935275862350837943562007745, −1.67747860100839052880721295462, −0.857586019376219966586882652585,
0.857586019376219966586882652585, 1.67747860100839052880721295462, 2.34935275862350837943562007745, 3.32154928460028163630088476789, 3.44686426377115101001934128516, 4.17049361604183668150248255904, 4.61153453491182141362175201622, 5.04533228961993992950876423947, 5.53658965585636024903423395160, 5.88908450915949561478136129218, 6.63762568339569193603820373666, 7.23852620954424289310886833172, 7.45985560667588965978165906144, 7.70041537652024622615801427573, 8.213624087600461730406930031285
