| L(s) = 1 | − 135.·2-s + 2.21e3·3-s + 1.02e4·4-s + 2.47e4·5-s − 3.00e5·6-s − 3.33e5·7-s − 2.86e5·8-s + 3.29e6·9-s − 3.36e6·10-s + 2.42e6·11-s + 2.27e7·12-s + 1.74e7·13-s + 4.53e7·14-s + 5.46e7·15-s − 4.54e7·16-s − 5.12e6·17-s − 4.47e8·18-s − 2.76e8·19-s + 2.54e8·20-s − 7.37e8·21-s − 3.30e8·22-s + 3.30e8·23-s − 6.33e8·24-s − 6.09e8·25-s − 2.37e9·26-s + 3.74e9·27-s − 3.43e9·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 1.75·3-s + 1.25·4-s + 0.707·5-s − 2.62·6-s − 1.07·7-s − 0.386·8-s + 2.06·9-s − 1.06·10-s + 0.413·11-s + 2.20·12-s + 1.00·13-s + 1.61·14-s + 1.23·15-s − 0.676·16-s − 0.0514·17-s − 3.10·18-s − 1.34·19-s + 0.889·20-s − 1.87·21-s − 0.620·22-s + 0.465·23-s − 0.676·24-s − 0.499·25-s − 1.50·26-s + 1.86·27-s − 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + 5.84e13T \) |
| good | 2 | \( 1 + 135.T + 8.19e3T^{2} \) |
| 3 | \( 1 - 2.21e3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 2.47e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.33e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 2.42e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.74e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 5.12e6T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.76e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 3.30e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.77e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 4.02e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 7.48e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.60e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.17e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 9.22e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.70e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 3.65e10T + 1.04e23T^{2} \) |
| 61 | \( 1 - 3.48e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 5.00e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 2.58e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 3.41e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 4.33e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.93e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 5.61e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 9.80e11T + 6.73e25T^{2} \) |
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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
−9.456715886295389332757133074047, −8.847506678988323179284605529018, −8.165236172763297691807641565161, −7.06511138100548203635723822167, −6.22808135960662817477308186030, −4.08728088760396877960793265512, −3.08772458981430647830178056362, −2.04926949358347143305661374446, −1.40389092243032014848496805273, 0,
1.40389092243032014848496805273, 2.04926949358347143305661374446, 3.08772458981430647830178056362, 4.08728088760396877960793265512, 6.22808135960662817477308186030, 7.06511138100548203635723822167, 8.165236172763297691807641565161, 8.847506678988323179284605529018, 9.456715886295389332757133074047
