| L(s) = 1 | + 14.7i·2-s + 243i·3-s + 1.83e3·4-s − 3.57e3·6-s + 7.99e4i·7-s + 5.71e4i·8-s − 5.90e4·9-s + 8.05e5·11-s + 4.45e5i·12-s + 1.19e6i·13-s − 1.17e6·14-s + 2.91e6·16-s + 2.63e6i·17-s − 8.69e5i·18-s − 1.16e7·19-s + ⋯ |
| L(s) = 1 | + 0.325i·2-s + 0.577i·3-s + 0.894·4-s − 0.187·6-s + 1.79i·7-s + 0.616i·8-s − 0.333·9-s + 1.50·11-s + 0.516i·12-s + 0.894i·13-s − 0.584·14-s + 0.693·16-s + 0.449i·17-s − 0.108i·18-s − 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.644961 + 2.73209i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.644961 + 2.73209i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 14.7iT - 2.04e3T^{2} \) |
| 7 | \( 1 - 7.99e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 - 8.05e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.19e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 - 2.63e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 + 1.16e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.84e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 1.90e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.01e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 8.06e7iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 2.26e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.67e9iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 8.58e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 - 3.52e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 4.35e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 1.65e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 7.58e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 2.75e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.22e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 2.43e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.20e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 + 4.44e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 2.04e10iT - 7.15e21T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.15632360225356901347155497868, −11.88389547973878138300597455132, −10.59765786081382904105356263800, −9.119545913405678128879446174817, −8.472232878309564065507304342459, −6.57994418812998748581155659262, −5.98415105949353363813956820952, −4.44194641141565809357522013916, −2.79236759944644210807983095685, −1.72991967101191072134478993511,
0.71366662398528229604792584127, 1.43033110611313781459876979066, 3.06182276315773274114416984147, 4.25094889481221188561230933559, 6.35360366903336033817774698864, 7.01138350820244042453617161598, 8.087035350159907845264294793394, 9.876258310944939222798095132109, 10.80589513851885289584057563158, 11.71639781602705563631795451860
