| L(s) = 1 | − 3·4-s − 8·7-s + 3·9-s − 6·13-s + 5·16-s − 6·17-s − 10·25-s + 24·28-s − 14·29-s − 9·36-s + 10·37-s − 4·43-s − 4·47-s + 34·49-s + 18·52-s − 53-s − 4·59-s − 24·63-s − 3·64-s + 18·68-s − 28·89-s + 48·91-s + 2·97-s + 30·100-s + 12·107-s − 40·112-s + 30·113-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 3.02·7-s + 9-s − 1.66·13-s + 5/4·16-s − 1.45·17-s − 2·25-s + 4.53·28-s − 2.59·29-s − 3/2·36-s + 1.64·37-s − 0.609·43-s − 0.583·47-s + 34/7·49-s + 2.49·52-s − 0.137·53-s − 0.520·59-s − 3.02·63-s − 3/8·64-s + 2.18·68-s − 2.96·89-s + 5.03·91-s + 0.203·97-s + 3·100-s + 1.16·107-s − 3.77·112-s + 2.82·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148877 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148877 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 53 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 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
−9.264238826193943461669666199356, −8.483283280441272019249165232631, −7.73762398215887034121172697318, −7.20736947999270954805514319970, −7.00110496145002859293965527187, −6.09508866331367665451434742179, −6.04347889941024738587947019148, −5.18496964971722877756379342667, −4.50862835068296220400083952922, −3.97664069548018217930915160992, −3.69121393392699379806557761029, −2.87754256168513552194439396359, −2.07674380795307242484164104235, 0, 0,
2.07674380795307242484164104235, 2.87754256168513552194439396359, 3.69121393392699379806557761029, 3.97664069548018217930915160992, 4.50862835068296220400083952922, 5.18496964971722877756379342667, 6.04347889941024738587947019148, 6.09508866331367665451434742179, 7.00110496145002859293965527187, 7.20736947999270954805514319970, 7.73762398215887034121172697318, 8.483283280441272019249165232631, 9.264238826193943461669666199356
