| L(s) = 1 | − 2·4-s − 2·5-s + 9-s − 8·13-s + 4·16-s + 16·17-s + 4·20-s − 7·25-s − 2·29-s − 2·36-s + 2·37-s − 20·41-s − 2·45-s − 5·49-s + 16·52-s + 20·53-s + 4·61-s − 8·64-s + 16·65-s − 32·68-s − 16·73-s − 8·80-s + 81-s − 32·85-s − 4·89-s + 10·97-s + 14·100-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s + 1/3·9-s − 2.21·13-s + 16-s + 3.88·17-s + 0.894·20-s − 7/5·25-s − 0.371·29-s − 1/3·36-s + 0.328·37-s − 3.12·41-s − 0.298·45-s − 5/7·49-s + 2.21·52-s + 2.74·53-s + 0.512·61-s − 64-s + 1.98·65-s − 3.88·68-s − 1.87·73-s − 0.894·80-s + 1/9·81-s − 3.47·85-s − 0.423·89-s + 1.01·97-s + 7/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 318096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 318096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 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
−8.405267645813230706380265272801, −8.018626195434666485016998626279, −7.67720525006885419220876512290, −7.30736849693508971465380420611, −7.04695512807619851939012088145, −5.83486052932247599921983260982, −5.67016112714498305999730280799, −5.07336814758302091908233894419, −4.75349830419579613526576050031, −4.01590674464272299745392011691, −3.36512175313045933289024112512, −3.31847810114563157193588872502, −2.11540780160413205543866999027, −1.09821079217653740897238739733, 0,
1.09821079217653740897238739733, 2.11540780160413205543866999027, 3.31847810114563157193588872502, 3.36512175313045933289024112512, 4.01590674464272299745392011691, 4.75349830419579613526576050031, 5.07336814758302091908233894419, 5.67016112714498305999730280799, 5.83486052932247599921983260982, 7.04695512807619851939012088145, 7.30736849693508971465380420611, 7.67720525006885419220876512290, 8.018626195434666485016998626279, 8.405267645813230706380265272801
