| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 3·8-s + 9-s + 10-s + 3·11-s + 12-s − 2·13-s − 2·14-s − 15-s + 16-s + 5·17-s − 18-s − 20-s + 2·21-s − 3·22-s + 2·23-s − 3·24-s − 2·25-s + 2·26-s + 27-s + 2·28-s − 7·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s − 0.554·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 0.223·20-s + 0.436·21-s − 0.639·22-s + 0.417·23-s − 0.612·24-s − 2/5·25-s + 0.392·26-s + 0.192·27-s + 0.377·28-s − 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5535 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5535 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7384982947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7384982947\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 52 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 50 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 62 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 208 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + T + 102 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−17.2858147064, −16.9930010744, −16.5147548082, −15.6538647405, −15.4277115023, −14.7191580717, −14.5873040137, −13.8980756593, −13.3112832684, −12.3378224341, −12.1231897724, −11.5465869042, −11.0853383860, −10.1814642814, −9.77673915210, −9.04365437581, −8.63346928016, −8.07545206138, −7.23562029799, −6.98905810761, −5.86021002683, −5.09694664683, −3.95672779865, −3.17630182360, −1.81712859369,
1.81712859369, 3.17630182360, 3.95672779865, 5.09694664683, 5.86021002683, 6.98905810761, 7.23562029799, 8.07545206138, 8.63346928016, 9.04365437581, 9.77673915210, 10.1814642814, 11.0853383860, 11.5465869042, 12.1231897724, 12.3378224341, 13.3112832684, 13.8980756593, 14.5873040137, 14.7191580717, 15.4277115023, 15.6538647405, 16.5147548082, 16.9930010744, 17.2858147064
