| L(s) = 1 | − 4·4-s − 36·11-s + 16·16-s − 46·19-s + 312·29-s − 170·31-s − 492·41-s + 144·44-s + 622·49-s + 1.58e3·59-s − 1.81e3·61-s − 64·64-s + 540·71-s + 184·76-s + 2.24e3·79-s − 396·89-s + 3.38e3·101-s − 1.18e3·109-s − 1.24e3·116-s − 1.69e3·121-s + 680·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.986·11-s + 1/4·16-s − 0.555·19-s + 1.99·29-s − 0.984·31-s − 1.87·41-s + 0.493·44-s + 1.81·49-s + 3.49·59-s − 3.80·61-s − 1/8·64-s + 0.902·71-s + 0.277·76-s + 3.19·79-s − 0.471·89-s + 3.33·101-s − 1.04·109-s − 0.998·116-s − 1.26·121-s + 0.492·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.675415825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675415825\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4330 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9601 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 23 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20365 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 156 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 85 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 p T + p^{3} T^{2} )( 1 + 12 p T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 122914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 124702 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 266425 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 792 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 907 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 497842 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 270 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 713518 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1123 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 549133 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 198 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 404482 T^{2} + p^{6} T^{4} \) |
| 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
−9.343993016173075581998105905121, −9.020044504121937035643701172503, −8.559179483849179370778482265961, −8.353195570458416279327667275473, −7.84753194929223053871921308921, −7.49155249123502089064396012275, −7.02802857816878580288347202770, −6.42862237274745087268140635416, −6.31849997047076611249966254867, −5.43890016358136146272405227830, −5.36677628565237915914982529739, −4.72326863637637469267147920416, −4.52675277112721968561828587942, −3.62930287688789995163077104701, −3.58054102821284159636179980814, −2.70578328638284405629365032981, −2.36217544100674672555370484987, −1.70784678278531129836639100425, −0.889540199155425317831500727959, −0.36897391655661560704941339429,
0.36897391655661560704941339429, 0.889540199155425317831500727959, 1.70784678278531129836639100425, 2.36217544100674672555370484987, 2.70578328638284405629365032981, 3.58054102821284159636179980814, 3.62930287688789995163077104701, 4.52675277112721968561828587942, 4.72326863637637469267147920416, 5.36677628565237915914982529739, 5.43890016358136146272405227830, 6.31849997047076611249966254867, 6.42862237274745087268140635416, 7.02802857816878580288347202770, 7.49155249123502089064396012275, 7.84753194929223053871921308921, 8.353195570458416279327667275473, 8.559179483849179370778482265961, 9.020044504121937035643701172503, 9.343993016173075581998105905121
