| L(s) = 1 | + 2-s + 3-s + 2·4-s − 5-s + 6-s + 5·8-s − 10-s + 4·11-s + 2·12-s − 4·13-s − 15-s + 5·16-s − 2·17-s − 4·19-s − 2·20-s + 4·22-s + 5·24-s − 4·26-s − 27-s − 4·29-s − 30-s + 10·32-s + 4·33-s − 2·34-s + 10·37-s − 4·38-s − 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.408·6-s + 1.76·8-s − 0.316·10-s + 1.20·11-s + 0.577·12-s − 1.10·13-s − 0.258·15-s + 5/4·16-s − 0.485·17-s − 0.917·19-s − 0.447·20-s + 0.852·22-s + 1.02·24-s − 0.784·26-s − 0.192·27-s − 0.742·29-s − 0.182·30-s + 1.76·32-s + 0.696·33-s − 0.342·34-s + 1.64·37-s − 0.648·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.483860302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.483860302\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−10.66223089414015292187910258194, −10.26461412756834315410193715547, −9.771775591936771372760003433854, −9.205401682070052884775321346239, −9.002585753001949999956752117096, −8.405168624582204226104281180940, −7.68115767985403750180112554286, −7.53395711904979411546516877796, −7.32085650447173629946201039853, −6.64936971076419900731175131158, −6.04441004913791683974173376665, −5.97892548508180626240841555998, −4.95777193820491724454535444046, −4.54436891368950025104062464127, −4.06898596196023554139454509089, −3.91123528636523196598119338496, −2.90790937860420428637668150658, −2.41799229872059879489546096535, −1.97272741241629886022960467241, −0.990280844832879975390847257813,
0.990280844832879975390847257813, 1.97272741241629886022960467241, 2.41799229872059879489546096535, 2.90790937860420428637668150658, 3.91123528636523196598119338496, 4.06898596196023554139454509089, 4.54436891368950025104062464127, 4.95777193820491724454535444046, 5.97892548508180626240841555998, 6.04441004913791683974173376665, 6.64936971076419900731175131158, 7.32085650447173629946201039853, 7.53395711904979411546516877796, 7.68115767985403750180112554286, 8.405168624582204226104281180940, 9.002585753001949999956752117096, 9.205401682070052884775321346239, 9.771775591936771372760003433854, 10.26461412756834315410193715547, 10.66223089414015292187910258194
