| L(s) = 1 | − 2-s − 4-s + 3·8-s + 2·9-s − 8·13-s − 16-s + 6·17-s − 2·18-s + 12·19-s + 2·25-s + 8·26-s − 5·32-s − 6·34-s − 2·36-s − 12·38-s + 8·43-s + 16·47-s + 49-s − 2·50-s + 8·52-s − 4·53-s − 12·59-s + 7·64-s − 8·67-s − 6·68-s + 6·72-s − 12·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s − 2.21·13-s − 1/4·16-s + 1.45·17-s − 0.471·18-s + 2.75·19-s + 2/5·25-s + 1.56·26-s − 0.883·32-s − 1.02·34-s − 1/3·36-s − 1.94·38-s + 1.21·43-s + 2.33·47-s + 1/7·49-s − 0.282·50-s + 1.10·52-s − 0.549·53-s − 1.56·59-s + 7/8·64-s − 0.977·67-s − 0.727·68-s + 0.707·72-s − 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067132886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067132886\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 142 T^{2} + p^{2} T^{4} \) |
| 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
−9.214495798386004607542201976347, −8.631079591423677880046838860916, −7.84933002885038219090820559705, −7.59058228874953958563134181143, −7.22917301408387071226050825214, −7.16590843720334965579082405488, −5.94482033122777382686029008782, −5.49802110261879426091744297525, −5.04859999650837657666486046486, −4.60481975306086236649689135333, −3.98148333284921969338329747901, −3.15767650340294705010056055923, −2.66353151165297197904172290622, −1.52509702395304645117002254448, −0.794120482552825607848750239033,
0.794120482552825607848750239033, 1.52509702395304645117002254448, 2.66353151165297197904172290622, 3.15767650340294705010056055923, 3.98148333284921969338329747901, 4.60481975306086236649689135333, 5.04859999650837657666486046486, 5.49802110261879426091744297525, 5.94482033122777382686029008782, 7.16590843720334965579082405488, 7.22917301408387071226050825214, 7.59058228874953958563134181143, 7.84933002885038219090820559705, 8.631079591423677880046838860916, 9.214495798386004607542201976347
