| L(s) = 1 | + 123.·2-s − 2.32e3·3-s − 1.75e4·4-s + 2.18e5·5-s − 2.87e5·6-s − 1.93e6·7-s − 6.20e6·8-s − 8.93e6·9-s + 2.69e7·10-s + 6.16e7·11-s + 4.07e7·12-s + 1.47e8·13-s − 2.39e8·14-s − 5.08e8·15-s − 1.92e8·16-s − 1.55e9·17-s − 1.10e9·18-s + 1.05e9·19-s − 3.83e9·20-s + 4.51e9·21-s + 7.61e9·22-s − 5.49e9·23-s + 1.44e10·24-s + 1.72e10·25-s + 1.82e10·26-s + 5.41e10·27-s + 3.39e10·28-s + ⋯ |
| L(s) = 1 | + 0.682·2-s − 0.614·3-s − 0.534·4-s + 1.25·5-s − 0.419·6-s − 0.889·7-s − 1.04·8-s − 0.622·9-s + 0.853·10-s + 0.953·11-s + 0.328·12-s + 0.651·13-s − 0.607·14-s − 0.768·15-s − 0.179·16-s − 0.921·17-s − 0.424·18-s + 0.270·19-s − 0.669·20-s + 0.546·21-s + 0.650·22-s − 0.336·23-s + 0.643·24-s + 0.566·25-s + 0.444·26-s + 0.996·27-s + 0.475·28-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
| \(L(8)\) |
\(\approx\) |
\(1.966524766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966524766\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 - 123.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 2.32e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 2.18e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 1.93e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 6.16e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.47e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.55e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 1.05e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 5.49e9T + 2.66e20T^{2} \) |
| 31 | \( 1 - 2.03e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 1.01e12T + 3.33e23T^{2} \) |
| 41 | \( 1 - 2.19e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.76e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 2.48e11T + 1.20e25T^{2} \) |
| 53 | \( 1 + 3.73e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 4.60e12T + 3.65e26T^{2} \) |
| 61 | \( 1 - 3.93e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 3.91e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.00e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 4.53e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.15e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 7.99e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 1.10e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 9.39e14T + 6.33e29T^{2} \) |
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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
−13.62799938373055870737762584845, −12.73880609373012299842356921333, −11.39310351580804322024263234821, −9.770258186128527038470567750510, −8.903915845989654908432575532224, −6.24844899417114193680851402635, −5.90500661166370974246236849276, −4.29542064372082994117541279798, −2.73179753903899880525771337096, −0.76964545063164973353121086068,
0.76964545063164973353121086068, 2.73179753903899880525771337096, 4.29542064372082994117541279798, 5.90500661166370974246236849276, 6.24844899417114193680851402635, 8.903915845989654908432575532224, 9.770258186128527038470567750510, 11.39310351580804322024263234821, 12.73880609373012299842356921333, 13.62799938373055870737762584845
