| L(s) = 1 | + 4-s + 2·7-s + 3·9-s − 4·11-s − 2·13-s + 16-s − 6·17-s − 9·25-s + 2·28-s + 4·29-s + 3·36-s + 6·37-s − 10·43-s − 4·44-s + 26·47-s − 11·49-s − 2·52-s + 12·53-s − 20·59-s + 6·63-s + 64-s − 6·68-s − 8·77-s + 12·89-s − 4·91-s + 28·97-s − 12·99-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 9-s − 1.20·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 9/5·25-s + 0.377·28-s + 0.742·29-s + 1/2·36-s + 0.986·37-s − 1.52·43-s − 0.603·44-s + 3.79·47-s − 1.57·49-s − 0.277·52-s + 1.64·53-s − 2.60·59-s + 0.755·63-s + 1/8·64-s − 0.727·68-s − 0.911·77-s + 1.27·89-s − 0.419·91-s + 2.84·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1898884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1898884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| 53 | $C_2$ | \( 1 - 12 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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
−7.45985560667588965978165906144, −7.44148850242498568563298207249, −6.71183737488766863530629500660, −6.33832588895369600957316852186, −5.88908450915949561478136129218, −5.40342951432596540306166728662, −4.78848689444739142342320835689, −4.61153453491182141362175201622, −4.09863386066861417262098583752, −3.55429641931114001585756761919, −2.74139103949267793883175763661, −2.27180100564839538823987009610, −1.97466976260941391586979329154, −1.13739318992545282171506597041, 0,
1.13739318992545282171506597041, 1.97466976260941391586979329154, 2.27180100564839538823987009610, 2.74139103949267793883175763661, 3.55429641931114001585756761919, 4.09863386066861417262098583752, 4.61153453491182141362175201622, 4.78848689444739142342320835689, 5.40342951432596540306166728662, 5.88908450915949561478136129218, 6.33832588895369600957316852186, 6.71183737488766863530629500660, 7.44148850242498568563298207249, 7.45985560667588965978165906144
