L(s) = 1 | − 18.8·2-s + 14.3·3-s + 228.·4-s + 382.·5-s − 270.·6-s + 343·7-s − 1.89e3·8-s − 1.98e3·9-s − 7.22e3·10-s − 3.04e3·11-s + 3.27e3·12-s − 2.19e3·13-s − 6.47e3·14-s + 5.48e3·15-s + 6.60e3·16-s − 1.13e4·17-s + 3.74e4·18-s + 4.23e3·19-s + 8.74e4·20-s + 4.91e3·21-s + 5.74e4·22-s − 3.14e4·23-s − 2.72e4·24-s + 6.83e4·25-s + 4.14e4·26-s − 5.97e4·27-s + 7.83e4·28-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 0.306·3-s + 1.78·4-s + 1.36·5-s − 0.511·6-s + 0.377·7-s − 1.31·8-s − 0.905·9-s − 2.28·10-s − 0.689·11-s + 0.547·12-s − 0.277·13-s − 0.630·14-s + 0.419·15-s + 0.402·16-s − 0.561·17-s + 1.51·18-s + 0.141·19-s + 2.44·20-s + 0.115·21-s + 1.15·22-s − 0.538·23-s − 0.402·24-s + 0.874·25-s + 0.462·26-s − 0.584·27-s + 0.674·28-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 | \( 1 + 18.8T + 128T^{2} \) |
| 3 | \( 1 - 14.3T + 2.18e3T^{2} \) |
| 5 | \( 1 - 382.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.04e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.13e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.23e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.81e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.19e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.69e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.63e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.96e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.11e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.45e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.90e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.29e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.73e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.50e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.85e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.74e6T + 8.07e13T^{2} \) |
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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
−11.69863384577301082886815260894, −10.58033919905253407248608371294, −9.816244431432426267984815279839, −8.851322928912059566982222366702, −8.039814656526592282399259559396, −6.65652516954608395110941999461, −5.36367246718780528503918142354, −2.61310674394530296847598498028, −1.69290717703264830958670281903, 0,
1.69290717703264830958670281903, 2.61310674394530296847598498028, 5.36367246718780528503918142354, 6.65652516954608395110941999461, 8.039814656526592282399259559396, 8.851322928912059566982222366702, 9.816244431432426267984815279839, 10.58033919905253407248608371294, 11.69863384577301082886815260894