| L(s) = 1 | + 2.08e4·2-s − 4.78e6·3-s − 1.00e8·4-s − 7.49e9·5-s − 9.98e10·6-s + 6.78e11·7-s − 1.33e13·8-s + 2.28e13·9-s − 1.56e14·10-s + 1.86e15·11-s + 4.82e14·12-s − 1.18e16·13-s + 1.41e16·14-s + 3.58e16·15-s − 2.23e17·16-s − 5.24e17·17-s + 4.77e17·18-s − 3.74e18·19-s + 7.55e17·20-s − 3.24e18·21-s + 3.90e19·22-s − 6.15e19·23-s + 6.36e19·24-s − 1.30e20·25-s − 2.47e20·26-s − 1.09e20·27-s − 6.83e19·28-s + ⋯ |
| L(s) = 1 | + 0.901·2-s − 0.577·3-s − 0.187·4-s − 0.549·5-s − 0.520·6-s + 0.377·7-s − 1.07·8-s + 0.333·9-s − 0.494·10-s + 1.48·11-s + 0.108·12-s − 0.834·13-s + 0.340·14-s + 0.317·15-s − 0.777·16-s − 0.755·17-s + 0.300·18-s − 1.07·19-s + 0.103·20-s − 0.218·21-s + 1.33·22-s − 1.10·23-s + 0.618·24-s − 0.698·25-s − 0.751·26-s − 0.192·27-s − 0.0709·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.448433042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.448433042\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4.78e6T \) |
| 7 | \( 1 - 6.78e11T \) |
| good | 2 | \( 1 - 2.08e4T + 5.36e8T^{2} \) |
| 5 | \( 1 + 7.49e9T + 1.86e20T^{2} \) |
| 11 | \( 1 - 1.86e15T + 1.58e30T^{2} \) |
| 13 | \( 1 + 1.18e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + 5.24e17T + 4.81e35T^{2} \) |
| 19 | \( 1 + 3.74e18T + 1.21e37T^{2} \) |
| 23 | \( 1 + 6.15e19T + 3.09e39T^{2} \) |
| 29 | \( 1 + 2.31e21T + 2.56e42T^{2} \) |
| 31 | \( 1 - 5.54e21T + 1.77e43T^{2} \) |
| 37 | \( 1 - 7.11e22T + 3.00e45T^{2} \) |
| 41 | \( 1 + 7.27e22T + 5.89e46T^{2} \) |
| 43 | \( 1 + 8.73e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 3.11e24T + 3.09e48T^{2} \) |
| 53 | \( 1 - 9.30e24T + 1.00e50T^{2} \) |
| 59 | \( 1 - 6.67e25T + 2.26e51T^{2} \) |
| 61 | \( 1 - 9.98e25T + 5.95e51T^{2} \) |
| 67 | \( 1 - 3.21e26T + 9.04e52T^{2} \) |
| 71 | \( 1 + 1.09e27T + 4.85e53T^{2} \) |
| 73 | \( 1 - 6.92e26T + 1.08e54T^{2} \) |
| 79 | \( 1 + 3.20e27T + 1.07e55T^{2} \) |
| 83 | \( 1 + 7.17e26T + 4.50e55T^{2} \) |
| 89 | \( 1 - 1.19e28T + 3.40e56T^{2} \) |
| 97 | \( 1 + 9.45e28T + 4.13e57T^{2} \) |
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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
−12.05028702983513171772525618176, −11.50550661954228209125181057888, −9.761540129889630847808284517954, −8.444344966766377199217090641002, −6.82900811916398113043497478157, −5.75111598162709043665232233093, −4.38840481827351231281155974356, −3.93674573252846255626781716272, −2.13100959655409350593330137099, −0.49929271559305763619656189077,
0.49929271559305763619656189077, 2.13100959655409350593330137099, 3.93674573252846255626781716272, 4.38840481827351231281155974356, 5.75111598162709043665232233093, 6.82900811916398113043497478157, 8.444344966766377199217090641002, 9.761540129889630847808284517954, 11.50550661954228209125181057888, 12.05028702983513171772525618176
