L(s) = 1 | − 11.5·2-s + 14.0·3-s + 5.35·4-s + 200.·5-s − 162.·6-s − 343·7-s + 1.41e3·8-s − 1.98e3·9-s − 2.31e3·10-s − 588.·11-s + 75.3·12-s + 2.19e3·13-s + 3.96e3·14-s + 2.82e3·15-s − 1.70e4·16-s + 2.00e4·17-s + 2.29e4·18-s − 9.98e3·19-s + 1.07e3·20-s − 4.82e3·21-s + 6.79e3·22-s + 4.22e4·23-s + 1.99e4·24-s − 3.78e4·25-s − 2.53e4·26-s − 5.87e4·27-s − 1.83e3·28-s + ⋯ |
L(s) = 1 | − 1.02·2-s + 0.300·3-s + 0.0418·4-s + 0.718·5-s − 0.307·6-s − 0.377·7-s + 0.978·8-s − 0.909·9-s − 0.733·10-s − 0.133·11-s + 0.0125·12-s + 0.277·13-s + 0.385·14-s + 0.216·15-s − 1.04·16-s + 0.987·17-s + 0.928·18-s − 0.333·19-s + 0.0300·20-s − 0.113·21-s + 0.136·22-s + 0.724·23-s + 0.294·24-s − 0.483·25-s − 0.283·26-s − 0.574·27-s − 0.0158·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
good | 2 | \( 1 + 11.5T + 128T^{2} \) |
| 3 | \( 1 - 14.0T + 2.18e3T^{2} \) |
| 5 | \( 1 - 200.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 588.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.00e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.98e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.22e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.63e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.26e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.00e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.80e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.79e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.63e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.90e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.41e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.97e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.36e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.10e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.27e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.81e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.02e7T + 8.07e13T^{2} \) |
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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
−12.03258981286826559777608033114, −10.59958441303215504960769220819, −9.819805660775348549509346203680, −8.828201808625923530539045632174, −8.020404989357962648987939144792, −6.50508599421838992184709863061, −5.10347929398800207091980518823, −3.15703091552755063106793971437, −1.54950365524627196511265655647, 0,
1.54950365524627196511265655647, 3.15703091552755063106793971437, 5.10347929398800207091980518823, 6.50508599421838992184709863061, 8.020404989357962648987939144792, 8.828201808625923530539045632174, 9.819805660775348549509346203680, 10.59958441303215504960769220819, 12.03258981286826559777608033114