| L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s + 4·11-s + 14-s + 16-s − 4·17-s + 2·20-s + 4·22-s − 23-s + 3·25-s + 28-s + 12·29-s − 4·31-s + 32-s − 4·34-s + 2·35-s + 4·37-s + 2·40-s − 6·41-s + 2·43-s + 4·44-s − 46-s − 47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.447·20-s + 0.852·22-s − 0.208·23-s + 3/5·25-s + 0.188·28-s + 2.22·29-s − 0.718·31-s + 0.176·32-s − 0.685·34-s + 0.338·35-s + 0.657·37-s + 0.316·40-s − 0.937·41-s + 0.304·43-s + 0.603·44-s − 0.147·46-s − 0.145·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.530435813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.530435813\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.5056209230, −13.3071566326, −12.5741355328, −12.1848918073, −11.9905051986, −11.3216484623, −11.0322770125, −10.5565430635, −10.0578246594, −9.63352098598, −9.12806079419, −8.67262338049, −8.29907560135, −7.59624245564, −7.01817725870, −6.48435797261, −6.32550024317, −5.71193763021, −5.06764337655, −4.54878050863, −4.18067112120, −3.34524040432, −2.69147413489, −1.95047608786, −1.24495904993,
1.24495904993, 1.95047608786, 2.69147413489, 3.34524040432, 4.18067112120, 4.54878050863, 5.06764337655, 5.71193763021, 6.32550024317, 6.48435797261, 7.01817725870, 7.59624245564, 8.29907560135, 8.67262338049, 9.12806079419, 9.63352098598, 10.0578246594, 10.5565430635, 11.0322770125, 11.3216484623, 11.9905051986, 12.1848918073, 12.5741355328, 13.3071566326, 13.5056209230
