| L(s) = 1 | + 4·3-s − 42·9-s − 84·11-s − 12·17-s − 188·19-s − 58·25-s − 292·27-s − 336·33-s + 108·41-s − 884·43-s + 82·49-s − 48·51-s − 752·57-s − 276·59-s + 356·67-s − 868·73-s − 232·75-s + 971·81-s − 540·83-s + 2.36e3·89-s − 2.47e3·97-s + 3.52e3·99-s + 2.60e3·107-s + 3.30e3·113-s + 2.63e3·121-s + 432·123-s + 127-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 1.55·9-s − 2.30·11-s − 0.171·17-s − 2.27·19-s − 0.463·25-s − 2.08·27-s − 1.77·33-s + 0.411·41-s − 3.13·43-s + 0.239·49-s − 0.131·51-s − 1.74·57-s − 0.609·59-s + 0.649·67-s − 1.39·73-s − 0.357·75-s + 1.33·81-s − 0.714·83-s + 2.81·89-s − 2.59·97-s + 3.58·99-s + 2.35·107-s + 2.74·113-s + 1.97·121-s + 0.316·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 58 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 82 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2666 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 94 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5134 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6710 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 47294 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 101114 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 442 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 204574 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 292954 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 138 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 269450 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 178 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 22226 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 434 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 958430 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1182 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1238 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30295338045405705545401047565, −10.80508703666155969517223378806, −10.41005255872427102154022004069, −10.00646001511202343791553557540, −9.317744818848529942745974686510, −8.662638531093527387411894040337, −8.351215401318940690732033937791, −8.186112224137385380868240675165, −7.58189839243084725143379158521, −6.90990366263998207737611360426, −6.04357930157275619139206792221, −5.87722611254869400186458400955, −5.03504245128758460320536565807, −4.67339428020119959120153714690, −3.61048924108128046570309144825, −3.08773135716927158656225292118, −2.37062466920971120401270890609, −2.06876675785090267017979841747, 0, 0,
2.06876675785090267017979841747, 2.37062466920971120401270890609, 3.08773135716927158656225292118, 3.61048924108128046570309144825, 4.67339428020119959120153714690, 5.03504245128758460320536565807, 5.87722611254869400186458400955, 6.04357930157275619139206792221, 6.90990366263998207737611360426, 7.58189839243084725143379158521, 8.186112224137385380868240675165, 8.351215401318940690732033937791, 8.662638531093527387411894040337, 9.317744818848529942745974686510, 10.00646001511202343791553557540, 10.41005255872427102154022004069, 10.80508703666155969517223378806, 11.30295338045405705545401047565
