| L(s) = 1 | − 3-s − 4-s + 4·5-s − 2·7-s + 9-s − 11-s + 12-s − 2·13-s − 4·15-s − 3·16-s + 4·17-s − 4·20-s + 2·21-s + 4·23-s + 6·25-s − 27-s + 2·28-s − 5·29-s − 9·31-s + 33-s − 8·35-s − 36-s + 7·37-s + 2·39-s − 7·41-s − 3·43-s + 44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1.78·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.554·13-s − 1.03·15-s − 3/4·16-s + 0.970·17-s − 0.894·20-s + 0.436·21-s + 0.834·23-s + 6/5·25-s − 0.192·27-s + 0.377·28-s − 0.928·29-s − 1.61·31-s + 0.174·33-s − 1.35·35-s − 1/6·36-s + 1.15·37-s + 0.320·39-s − 1.09·41-s − 0.457·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6837807667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6837807667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 40 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 128 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T - 68 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 116 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−18.1273053191, −17.5822274612, −17.0408017565, −16.6346146257, −16.3283210215, −15.4776343788, −14.7588633018, −14.3805105409, −13.6458603680, −13.1437876027, −12.9879482895, −12.3114485954, −11.4271730929, −10.8772211459, −10.0348078402, −9.69114917517, −9.39745415616, −8.58076353066, −7.47935107360, −6.79106928034, −6.05629532553, −5.42994248504, −4.88946664500, −3.47266737166, −2.09904525147,
2.09904525147, 3.47266737166, 4.88946664500, 5.42994248504, 6.05629532553, 6.79106928034, 7.47935107360, 8.58076353066, 9.39745415616, 9.69114917517, 10.0348078402, 10.8772211459, 11.4271730929, 12.3114485954, 12.9879482895, 13.1437876027, 13.6458603680, 14.3805105409, 14.7588633018, 15.4776343788, 16.3283210215, 16.6346146257, 17.0408017565, 17.5822274612, 18.1273053191
