| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s + 16.5·5-s + 216·6-s + 343·7-s + 512·8-s + 729·9-s + 132.·10-s − 2.50e3·11-s + 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s + 447.·15-s + 4.09e3·16-s + 8.55e3·17-s + 5.83e3·18-s − 2.26e4·19-s + 1.06e3·20-s + 9.26e3·21-s − 2.00e4·22-s − 8.69e4·23-s + 1.38e4·24-s − 7.78e4·25-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 + 0.0593·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.0419·10-s − 0.568·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.0342·15-s + 0.250·16-s + 0.422·17-s + 0.235·18-s − 0.759·19-s + 0.0296·20-s + 0.218·21-s − 0.402·22-s − 1.48·23-s + 0.204·24-s − 0.996·25-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 16.5T + 7.81e4T^{2} \) |
| 11 | \( 1 + 2.50e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 8.55e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.26e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.69e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.64e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.67e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.51e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.13e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.40e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.21e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.83e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.00e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.55e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.62e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.02e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.81e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.51e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.70e6T + 8.07e13T^{2} \) |
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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
−9.226367802977069740654875663712, −8.096827743081410306252803350492, −7.57345931900643256240480910435, −6.38337535251010365409603774691, −5.45316071589083401357065010514, −4.44645071339704296204032260652, −3.56665870064871692487906485393, −2.43428744318733589074823890131, −1.64556182205884265481490612431, 0,
1.64556182205884265481490612431, 2.43428744318733589074823890131, 3.56665870064871692487906485393, 4.44645071339704296204032260652, 5.45316071589083401357065010514, 6.38337535251010365409603774691, 7.57345931900643256240480910435, 8.096827743081410306252803350492, 9.226367802977069740654875663712
