L(s) = 1 | − 24·3-s + 333·9-s − 1.29e3·11-s + 484·13-s + 2.59e3·23-s − 86·25-s − 2.16e3·27-s + 3.11e4·33-s + 2.41e4·37-s − 1.16e4·39-s + 2.59e4·47-s + 1.53e3·49-s − 1.68e4·59-s + 5.15e4·61-s − 6.22e4·69-s + 1.11e5·71-s + 5.20e4·73-s + 2.06e3·75-s − 2.90e4·81-s + 1.56e5·83-s + 2.06e5·97-s − 4.31e5·99-s − 4.27e4·107-s − 1.17e5·109-s − 5.78e5·111-s + 1.61e5·117-s + 9.37e5·121-s + ⋯ |
L(s) = 1 | − 1.53·3-s + 1.37·9-s − 3.22·11-s + 0.794·13-s + 1.02·23-s − 0.0275·25-s − 0.570·27-s + 4.97·33-s + 2.89·37-s − 1.22·39-s + 1.71·47-s + 0.0915·49-s − 0.630·59-s + 1.77·61-s − 1.57·69-s + 2.62·71-s + 1.14·73-s + 0.0423·75-s − 0.492·81-s + 2.49·83-s + 2.22·97-s − 4.42·99-s − 0.361·107-s − 0.949·109-s − 4.45·111-s + 1.08·117-s + 5.82·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.286786502\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.286786502\) |
\(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 8 p T + p^{5} T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 86 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1538 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 648 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 242 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2738338 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3380474 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 1296 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 37062298 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 45678866 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12058 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12512146 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 33190390 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12960 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 108251530 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8424 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 25762 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2596035290 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 55728 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 26026 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 5786874674 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 78408 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4075828594 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 103090 T + p^{5} T^{2} )^{2} \) |
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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
−11.75537587883097289521548577003, −11.23295619784704986814143865368, −10.92180859889933994962978113216, −10.58886977688223684302470430769, −10.21883363297717837740418648208, −9.589261765366119003100241709465, −8.959297964935179484909070059051, −7.994885252790407015107938060628, −7.943323719060211160038269363593, −7.31990254292041021029335593226, −6.52190530222904533827431060957, −6.08654065435029716608285973120, −5.34076479752750211488141309264, −5.23922539144430193406757194369, −4.63234138595863434178496289738, −3.74722332757526138354548903679, −2.71192422636349833860785161696, −2.30576158348564855887970448708, −0.807998378110374456120912133353, −0.57002883313512423961339566119,
0.57002883313512423961339566119, 0.807998378110374456120912133353, 2.30576158348564855887970448708, 2.71192422636349833860785161696, 3.74722332757526138354548903679, 4.63234138595863434178496289738, 5.23922539144430193406757194369, 5.34076479752750211488141309264, 6.08654065435029716608285973120, 6.52190530222904533827431060957, 7.31990254292041021029335593226, 7.943323719060211160038269363593, 7.994885252790407015107938060628, 8.959297964935179484909070059051, 9.589261765366119003100241709465, 10.21883363297717837740418648208, 10.58886977688223684302470430769, 10.92180859889933994962978113216, 11.23295619784704986814143865368, 11.75537587883097289521548577003