L(s) = 1 | − 19.7·2-s − 27·3-s + 261.·4-s − 90.5·5-s + 532.·6-s − 807.·7-s − 2.63e3·8-s + 729·9-s + 1.78e3·10-s + 4.97e3·11-s − 7.06e3·12-s − 2.92e3·13-s + 1.59e4·14-s + 2.44e3·15-s + 1.85e4·16-s − 2.93e4·17-s − 1.43e4·18-s + 8.90e3·19-s − 2.36e4·20-s + 2.18e4·21-s − 9.81e4·22-s − 2.13e4·23-s + 7.11e4·24-s − 6.99e4·25-s + 5.76e4·26-s − 1.96e4·27-s − 2.11e5·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s − 0.577·3-s + 2.04·4-s − 0.324·5-s + 1.00·6-s − 0.890·7-s − 1.81·8-s + 0.333·9-s + 0.565·10-s + 1.12·11-s − 1.17·12-s − 0.368·13-s + 1.55·14-s + 0.187·15-s + 1.13·16-s − 1.44·17-s − 0.581·18-s + 0.297·19-s − 0.662·20-s + 0.513·21-s − 1.96·22-s − 0.365·23-s + 1.05·24-s − 0.894·25-s + 0.643·26-s − 0.192·27-s − 1.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
good | 2 | \( 1 + 19.7T + 128T^{2} \) |
| 5 | \( 1 + 90.5T + 7.81e4T^{2} \) |
| 7 | \( 1 + 807.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.93e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 8.90e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.13e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.44e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.24e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.49e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.69e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.14e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.91e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.23e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.01e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.76e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.82e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.01e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.38e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.52e7T + 8.07e13T^{2} \) |
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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
−10.71042576315969463079689493879, −9.693924085864529131754711145945, −9.113769986859210026588758178488, −7.924951495822583243516048459259, −6.81845557333484972922038763169, −6.22405931252589715892827833297, −4.19753475606811314431755752728, −2.45414143509716166340650843244, −0.990043362036185300131713793310, 0,
0.990043362036185300131713793310, 2.45414143509716166340650843244, 4.19753475606811314431755752728, 6.22405931252589715892827833297, 6.81845557333484972922038763169, 7.924951495822583243516048459259, 9.113769986859210026588758178488, 9.693924085864529131754711145945, 10.71042576315969463079689493879