| L(s) = 1 | + 1.22·2-s − 1.15·3-s − 6.50·4-s − 5·5-s − 1.41·6-s + 14.6·7-s − 17.7·8-s − 25.6·9-s − 6.10·10-s − 71.3·11-s + 7.53·12-s − 28.4·13-s + 17.9·14-s + 5.78·15-s + 30.4·16-s + 17·17-s − 31.3·18-s + 85.9·19-s + 32.5·20-s − 16.9·21-s − 87.1·22-s − 7.17·23-s + 20.5·24-s + 25·25-s − 34.6·26-s + 60.9·27-s − 95.4·28-s + ⋯ |
| L(s) = 1 | + 0.431·2-s − 0.222·3-s − 0.813·4-s − 0.447·5-s − 0.0962·6-s + 0.792·7-s − 0.783·8-s − 0.950·9-s − 0.193·10-s − 1.95·11-s + 0.181·12-s − 0.605·13-s + 0.342·14-s + 0.0996·15-s + 0.475·16-s + 0.242·17-s − 0.410·18-s + 1.03·19-s + 0.363·20-s − 0.176·21-s − 0.845·22-s − 0.0650·23-s + 0.174·24-s + 0.200·25-s − 0.261·26-s + 0.434·27-s − 0.644·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 1.22T + 8T^{2} \) |
| 3 | \( 1 + 1.15T + 27T^{2} \) |
| 7 | \( 1 - 14.6T + 343T^{2} \) |
| 11 | \( 1 + 71.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.4T + 2.19e3T^{2} \) |
| 19 | \( 1 - 85.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.17T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 15.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 67.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 38.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 251.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 57.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 677.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 598.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 346.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 849.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 911.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 704.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 55.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 72.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 972.T + 9.12e5T^{2} \) |
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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
−13.27861730112758605271410278167, −12.20926072075997838797404750033, −11.21072371738489772343609553653, −9.968692888679471408754258890624, −8.482916693478330033644212279507, −7.67882372524321502275345666771, −5.51473142833489296868824453609, −4.87101711243109012476822227500, −3.02655369819734180346549618816, 0,
3.02655369819734180346549618816, 4.87101711243109012476822227500, 5.51473142833489296868824453609, 7.67882372524321502275345666771, 8.482916693478330033644212279507, 9.968692888679471408754258890624, 11.21072371738489772343609553653, 12.20926072075997838797404750033, 13.27861730112758605271410278167
