L(s) = 1 | + 223·4-s − 6.56e3·9-s − 1.58e4·16-s + 4.07e5·19-s + 3.66e6·31-s − 1.46e6·36-s − 1.15e7·49-s + 2.86e7·61-s − 1.81e7·64-s + 9.09e7·76-s + 1.39e8·79-s + 4.30e7·81-s − 5.19e8·109-s + 4.28e8·121-s + 8.16e8·124-s + 127-s + 131-s + 137-s + 139-s + 1.03e8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.63e9·169-s − 2.67e9·171-s + ⋯ |
L(s) = 1 | + 0.871·4-s − 9-s − 0.241·16-s + 3.13·19-s + 3.96·31-s − 0.871·36-s − 2·49-s + 2.06·61-s − 1.08·64-s + 2.72·76-s + 3.57·79-s + 81-s − 3.67·109-s + 2·121-s + 3.45·124-s + 0.241·144-s − 2·169-s − 3.13·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(3.860452937\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.860452937\) |
\(L(5)\) |
|
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 | $C_2$ | \( 1 + p^{8} T^{2} \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 223 T^{2} + p^{16} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 13505544958 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 203998 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 145963835522 T^{2} + p^{16} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1831682 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 17436524386562 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34736892000962 T^{2} + p^{16} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14324642 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 69617278 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4489783501049278 T^{2} + p^{16} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
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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
−13.54420653410565103323398183751, −12.20709965402453296629993409195, −12.03463020386680539693085394798, −11.42103550288830668724248085958, −11.28625091630429291855988155624, −10.36121985185948969219155767194, −9.712297620810837928519282221089, −9.425374975517169057412823621140, −8.374432549960790948234631982260, −8.020341811852842834617252662091, −7.35988039820324632836249891454, −6.54040448354097979665897131091, −6.24385332469594188633991352750, −5.22683797328678044307727658321, −4.88738845624000181590951079640, −3.60242279634992969322859654286, −2.88730279620532147633317227312, −2.49637904898507429020512654708, −1.26365004012369579093150663173, −0.66897873447030978731638254418,
0.66897873447030978731638254418, 1.26365004012369579093150663173, 2.49637904898507429020512654708, 2.88730279620532147633317227312, 3.60242279634992969322859654286, 4.88738845624000181590951079640, 5.22683797328678044307727658321, 6.24385332469594188633991352750, 6.54040448354097979665897131091, 7.35988039820324632836249891454, 8.020341811852842834617252662091, 8.374432549960790948234631982260, 9.425374975517169057412823621140, 9.712297620810837928519282221089, 10.36121985185948969219155767194, 11.28625091630429291855988155624, 11.42103550288830668724248085958, 12.03463020386680539693085394798, 12.20709965402453296629993409195, 13.54420653410565103323398183751