| L(s) = 1 | + 27.7·2-s + 192.·3-s + 259.·4-s − 1.97e3·5-s + 5.35e3·6-s + 2.40e3·7-s − 7.01e3·8-s + 1.75e4·9-s − 5.49e4·10-s − 6.44e4·11-s + 5.00e4·12-s + 2.85e4·13-s + 6.66e4·14-s − 3.81e5·15-s − 3.27e5·16-s − 7.36e4·17-s + 4.87e5·18-s + 4.78e5·19-s − 5.13e5·20-s + 4.63e5·21-s − 1.79e6·22-s − 1.93e6·23-s − 1.35e6·24-s + 1.96e6·25-s + 7.93e5·26-s − 4.13e5·27-s + 6.23e5·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 1.37·3-s + 0.506·4-s − 1.41·5-s + 1.68·6-s + 0.377·7-s − 0.605·8-s + 0.891·9-s − 1.73·10-s − 1.32·11-s + 0.697·12-s + 0.277·13-s + 0.463·14-s − 1.94·15-s − 1.24·16-s − 0.213·17-s + 1.09·18-s + 0.842·19-s − 0.717·20-s + 0.519·21-s − 1.63·22-s − 1.44·23-s − 0.832·24-s + 1.00·25-s + 0.340·26-s − 0.149·27-s + 0.191·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 - 27.7T + 512T^{2} \) |
| 3 | \( 1 - 192.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.97e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 6.44e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 7.36e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 4.78e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.93e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.36e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.84e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.08e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.19e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.57e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.07e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.39e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.31e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.50e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.39e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.79e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.99e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.07e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.00e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.96e8T + 7.60e17T^{2} \) |
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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
−12.05842209319255636056734765079, −10.97073232228727492953436032917, −9.248742879019446993506412689529, −8.097920722317711918552529244712, −7.49556640301794637512032788026, −5.50155986518860489600902685326, −4.16357146227950475674860914482, −3.46504364421280210551555014798, −2.36117377561303802496071297074, 0,
2.36117377561303802496071297074, 3.46504364421280210551555014798, 4.16357146227950475674860914482, 5.50155986518860489600902685326, 7.49556640301794637512032788026, 8.097920722317711918552529244712, 9.248742879019446993506412689529, 10.97073232228727492953436032917, 12.05842209319255636056734765079
