| L(s) = 1 | + 3.76·2-s + 9.31·3-s + 6.17·4-s + 1.25·5-s + 35.0·6-s + 7·7-s − 6.86·8-s + 59.7·9-s + 4.71·10-s − 42.4·11-s + 57.5·12-s − 78.8·13-s + 26.3·14-s + 11.6·15-s − 75.2·16-s + 118.·17-s + 224.·18-s + 2.87·19-s + 7.74·20-s + 65.1·21-s − 159.·22-s + 23·23-s − 63.9·24-s − 123.·25-s − 296.·26-s + 304.·27-s + 43.2·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 1.79·3-s + 0.772·4-s + 0.112·5-s + 2.38·6-s + 0.377·7-s − 0.303·8-s + 2.21·9-s + 0.149·10-s − 1.16·11-s + 1.38·12-s − 1.68·13-s + 0.503·14-s + 0.200·15-s − 1.17·16-s + 1.69·17-s + 2.94·18-s + 0.0346·19-s + 0.0865·20-s + 0.677·21-s − 1.54·22-s + 0.208·23-s − 0.543·24-s − 0.987·25-s − 2.23·26-s + 2.17·27-s + 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.226820253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.226820253\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 3.76T + 8T^{2} \) |
| 3 | \( 1 - 9.31T + 27T^{2} \) |
| 5 | \( 1 - 1.25T + 125T^{2} \) |
| 11 | \( 1 + 42.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.87T + 6.85e3T^{2} \) |
| 29 | \( 1 + 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 389.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 286.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 103.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 515.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 424.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 631.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 365.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 285.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 531.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 151.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 115.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 979.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 940.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 688.T + 9.12e5T^{2} \) |
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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
−12.75726716869030014651025109462, −11.97790980739291257161871964174, −10.16506196423058432868857622511, −9.395207519982842115764360626599, −8.002825970027637702388691734173, −7.39569403323221647441732035503, −5.49838093473923829956289490153, −4.42339533367375514170303391184, −3.14390286796507210575197678730, −2.32371799196226747716281917270,
2.32371799196226747716281917270, 3.14390286796507210575197678730, 4.42339533367375514170303391184, 5.49838093473923829956289490153, 7.39569403323221647441732035503, 8.002825970027637702388691734173, 9.395207519982842115764360626599, 10.16506196423058432868857622511, 11.97790980739291257161871964174, 12.75726716869030014651025109462
