| L(s) = 1 | − 288.·2-s + 2.29e3·3-s + 5.04e4·4-s − 1.30e5·5-s − 6.61e5·6-s − 2.37e6·7-s − 5.08e6·8-s − 9.09e6·9-s + 3.75e7·10-s − 6.52e7·11-s + 1.15e8·12-s + 1.63e8·13-s + 6.83e8·14-s − 2.98e8·15-s − 1.84e8·16-s + 2.09e9·17-s + 2.62e9·18-s − 5.84e9·19-s − 6.55e9·20-s − 5.43e9·21-s + 1.88e10·22-s − 2.93e10·23-s − 1.16e10·24-s − 1.36e10·25-s − 4.70e10·26-s − 5.37e10·27-s − 1.19e11·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 0.605·3-s + 1.53·4-s − 0.744·5-s − 0.964·6-s − 1.08·7-s − 0.858·8-s − 0.633·9-s + 1.18·10-s − 1.00·11-s + 0.931·12-s + 0.721·13-s + 1.73·14-s − 0.450·15-s − 0.171·16-s + 1.23·17-s + 1.00·18-s − 1.50·19-s − 1.14·20-s − 0.658·21-s + 1.60·22-s − 1.79·23-s − 0.519·24-s − 0.445·25-s − 1.14·26-s − 0.988·27-s − 1.67·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.3055425519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3055425519\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 1.72e10T \) |
| good | 2 | \( 1 + 288.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 2.29e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 1.30e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 2.37e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 6.52e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.63e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.09e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 5.84e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.93e10T + 2.66e20T^{2} \) |
| 31 | \( 1 - 1.68e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 2.70e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 2.20e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 3.01e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 1.96e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 3.23e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 2.03e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 1.45e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 1.66e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 6.64e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.21e14T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.50e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.23e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 1.48e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 4.67e14T + 6.33e29T^{2} \) |
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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
−13.72732824249202622683353435737, −12.16321225860713674554806381053, −10.70099283813459975402728411039, −9.722733023246099108591321139477, −8.394279080559413712349789791610, −7.84327314187775589806703764222, −6.20389830455531631546156571535, −3.59322988815852906646223544453, −2.25043373862357561265521557710, −0.37416273673151056489617648997,
0.37416273673151056489617648997, 2.25043373862357561265521557710, 3.59322988815852906646223544453, 6.20389830455531631546156571535, 7.84327314187775589806703764222, 8.394279080559413712349789791610, 9.722733023246099108591321139477, 10.70099283813459975402728411039, 12.16321225860713674554806381053, 13.72732824249202622683353435737
