| L(s) = 1 | − 30.8·2-s − 107.·3-s + 436.·4-s − 558.·5-s + 3.30e3·6-s − 8.31e3·7-s + 2.31e3·8-s − 8.15e3·9-s + 1.72e4·10-s − 7.71e4·11-s − 4.68e4·12-s + 8.18e4·13-s + 2.56e5·14-s + 5.99e4·15-s − 2.95e5·16-s + 3.39e5·17-s + 2.51e5·18-s − 2.59e5·19-s − 2.43e5·20-s + 8.93e5·21-s + 2.37e6·22-s − 1.85e6·23-s − 2.48e5·24-s − 1.64e6·25-s − 2.52e6·26-s + 2.98e6·27-s − 3.63e6·28-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 0.765·3-s + 0.852·4-s − 0.399·5-s + 1.04·6-s − 1.30·7-s + 0.200·8-s − 0.414·9-s + 0.543·10-s − 1.58·11-s − 0.652·12-s + 0.794·13-s + 1.78·14-s + 0.305·15-s − 1.12·16-s + 0.985·17-s + 0.564·18-s − 0.456·19-s − 0.340·20-s + 1.00·21-s + 2.16·22-s − 1.38·23-s − 0.153·24-s − 0.840·25-s − 1.08·26-s + 1.08·27-s − 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 197 | \( 1 + 1.50e9T \) |
| good | 2 | \( 1 + 30.8T + 512T^{2} \) |
| 3 | \( 1 + 107.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 558.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.31e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.71e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 8.18e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.39e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.59e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.85e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.16e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.31e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.47e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.19e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.45e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.11e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.99e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.17e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 7.09e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.92e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.86e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.03e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 8.76e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.35e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.85e7T + 7.60e17T^{2} \) |
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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
−10.34639653111310761907760832172, −9.467295346442820652243553812983, −8.306141669227629049040610539952, −7.60108101177536351673389669603, −6.32820807619317909864552309031, −5.45242451403417197636401435643, −3.74776290956352335795872136456, −2.36971915770509447229520612779, −0.66262904691575939657754896534, 0,
0.66262904691575939657754896534, 2.36971915770509447229520612779, 3.74776290956352335795872136456, 5.45242451403417197636401435643, 6.32820807619317909864552309031, 7.60108101177536351673389669603, 8.306141669227629049040610539952, 9.467295346442820652243553812983, 10.34639653111310761907760832172
