| L(s) = 1 | − 6·3-s − 4·4-s − 2·5-s + 21·9-s − 11-s + 24·12-s + 12·15-s + 12·16-s + 8·20-s − 10·23-s − 7·25-s − 54·27-s + 2·31-s + 6·33-s − 84·36-s − 10·37-s + 4·44-s − 42·45-s + 16·47-s − 72·48-s + 49-s − 12·53-s + 2·55-s + 6·59-s − 48·60-s − 32·64-s − 6·67-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 2·4-s − 0.894·5-s + 7·9-s − 0.301·11-s + 6.92·12-s + 3.09·15-s + 3·16-s + 1.78·20-s − 2.08·23-s − 7/5·25-s − 10.3·27-s + 0.359·31-s + 1.04·33-s − 14·36-s − 1.64·37-s + 0.603·44-s − 6.26·45-s + 2.33·47-s − 10.3·48-s + 1/7·49-s − 1.64·53-s + 0.269·55-s + 0.781·59-s − 6.19·60-s − 4·64-s − 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 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
−9.842751379499007423618867754387, −9.057617126467774810255940907586, −8.090125574816450530034167252480, −7.956327480837723148155835866573, −7.17057178520211720164419084101, −6.56709274805536909588701134945, −5.80118601537640384820432803021, −5.58596676881303008604275522263, −5.30170245060560361736841457140, −4.39514744576497898862871693801, −4.26020787258843561459488122037, −3.79024025385094143595808125256, −1.32889476078570135837429482480, 0, 0,
1.32889476078570135837429482480, 3.79024025385094143595808125256, 4.26020787258843561459488122037, 4.39514744576497898862871693801, 5.30170245060560361736841457140, 5.58596676881303008604275522263, 5.80118601537640384820432803021, 6.56709274805536909588701134945, 7.17057178520211720164419084101, 7.956327480837723148155835866573, 8.090125574816450530034167252480, 9.057617126467774810255940907586, 9.842751379499007423618867754387
