| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 330.·5-s + 216·6-s − 343·7-s + 512·8-s + 729·9-s − 2.64e3·10-s + 4.51e3·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s − 8.93e3·15-s + 4.09e3·16-s + 1.39e4·17-s + 5.83e3·18-s − 1.79e4·19-s − 2.11e4·20-s − 9.26e3·21-s + 3.61e4·22-s − 1.32e4·23-s + 1.38e4·24-s + 3.13e4·25-s − 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.18·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.837·10-s + 1.02·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.683·15-s + 0.250·16-s + 0.687·17-s + 0.235·18-s − 0.600·19-s − 0.591·20-s − 0.218·21-s + 0.723·22-s − 0.226·23-s + 0.204·24-s + 0.401·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.499748102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.499748102\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 5 | \( 1 + 330.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.51e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.39e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.32e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.17e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.72e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.48e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.21e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.24e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.00e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.68e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.00e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.11e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.03e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.41e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.58e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.09e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.32e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.39e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.26e7T + 8.07e13T^{2} \) |
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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
−9.629592895374294728891496047527, −8.665206081487724332404276278635, −7.76833274572432007167128023764, −7.02592470818549452510388581451, −6.08419543547605186359088981236, −4.74326346100393078366818513803, −3.85134695240561572943516035617, −3.32491062513298131963994493722, −2.04560693895195496766410923942, −0.69886990583375031604937258044,
0.69886990583375031604937258044, 2.04560693895195496766410923942, 3.32491062513298131963994493722, 3.85134695240561572943516035617, 4.74326346100393078366818513803, 6.08419543547605186359088981236, 7.02592470818549452510388581451, 7.76833274572432007167128023764, 8.665206081487724332404276278635, 9.629592895374294728891496047527
