| L(s) = 1 | + 83.0·5-s + 37.1·7-s − 1.95e3·11-s − 4.30e3i·13-s − 7.85e3i·17-s − 6.11e3i·19-s + 1.55e4i·23-s − 8.72e3·25-s + 1.66e3·29-s + 2.78e4·31-s + 3.08e3·35-s + 5.23e4i·37-s + 5.18e4i·41-s + 1.35e5i·43-s − 9.36e4i·47-s + ⋯ |
| L(s) = 1 | + 0.664·5-s + 0.108·7-s − 1.47·11-s − 1.96i·13-s − 1.59i·17-s − 0.891i·19-s + 1.27i·23-s − 0.558·25-s + 0.0684·29-s + 0.935·31-s + 0.0719·35-s + 1.03i·37-s + 0.751i·41-s + 1.70i·43-s − 0.901i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.639i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.04446813737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04446813737\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 83.0T + 1.56e4T^{2} \) |
| 7 | \( 1 - 37.1T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.95e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 4.30e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 7.85e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 6.11e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.55e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.66e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.78e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 5.23e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 5.18e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.35e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 9.36e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.04e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.76e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 926. iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.63e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 6.10e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.33e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 6.05e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 4.86e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 8.26e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.11e6T + 8.32e11T^{2} \) |
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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
−9.967322238916623720633348256407, −9.520265020841315588398099875715, −8.069854754359530584351212421116, −7.71213173401797077605088935823, −6.43142722768413120762786746579, −5.29044273488767357471980324649, −4.99479831525798020548694005643, −3.09291244920408094451444699978, −2.59858378814174331653415404547, −1.04975492183478999162758552928,
0.008870339705052185557399888798, 1.70856862622770898770365016521, 2.30230165737896016605590156254, 3.82072782115606864071652083625, 4.76070707280011006139210820312, 5.88961271763456668856323029045, 6.53730615324425798742130188024, 7.75122890925643302754971735954, 8.534975563116917007312287338995, 9.466806735265060470760727660555
