| L(s) = 1 | + 4-s − 5-s + 9-s − 4·11-s − 20-s + 36-s − 4·44-s − 45-s + 2·49-s + 4·55-s + 2·61-s − 64-s − 4·99-s + 2·101-s + 10·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s + 9-s − 4·11-s − 20-s + 36-s − 4·44-s − 45-s + 2·49-s + 4·55-s + 2·61-s − 64-s − 4·99-s + 2·101-s + 10·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3258025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8193418453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8193418453\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
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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
−10.10194378396398369010649243791, −9.284233658200396951443201340477, −8.662222920761846503929618367783, −8.343703000118279132387933579893, −7.902099249505173703172214692599, −7.59717800370047158160619429912, −7.52261717655196593223978452712, −6.89272211029900109895360237539, −6.84912811141498521671640861474, −5.78470530837588323895847250057, −5.76501086707982901293560452799, −5.19030032299635478293281023225, −4.77737323369318287143758728101, −4.37573129061761763166196690533, −3.80210848357441750356037224391, −3.15057363362349909747451420927, −2.77320480571584977700272445276, −2.25804372038258706175905864861, −1.97808180144457881941174612259, −0.63738001368318639160124004468,
0.63738001368318639160124004468, 1.97808180144457881941174612259, 2.25804372038258706175905864861, 2.77320480571584977700272445276, 3.15057363362349909747451420927, 3.80210848357441750356037224391, 4.37573129061761763166196690533, 4.77737323369318287143758728101, 5.19030032299635478293281023225, 5.76501086707982901293560452799, 5.78470530837588323895847250057, 6.84912811141498521671640861474, 6.89272211029900109895360237539, 7.52261717655196593223978452712, 7.59717800370047158160619429912, 7.902099249505173703172214692599, 8.343703000118279132387933579893, 8.662222920761846503929618367783, 9.284233658200396951443201340477, 10.10194378396398369010649243791
