| L(s) = 1 | − 1.81·2-s + 6.14·3-s − 4.69·4-s − 5·5-s − 11.1·6-s − 34.0·7-s + 23.0·8-s + 10.8·9-s + 9.08·10-s − 33.6·11-s − 28.8·12-s + 25.1·13-s + 61.8·14-s − 30.7·15-s − 4.37·16-s + 17·17-s − 19.6·18-s − 150.·19-s + 23.4·20-s − 209.·21-s + 61.2·22-s − 126.·23-s + 141.·24-s + 25·25-s − 45.6·26-s − 99.5·27-s + 159.·28-s + ⋯ |
| L(s) = 1 | − 0.642·2-s + 1.18·3-s − 0.587·4-s − 0.447·5-s − 0.760·6-s − 1.83·7-s + 1.01·8-s + 0.400·9-s + 0.287·10-s − 0.923·11-s − 0.694·12-s + 0.536·13-s + 1.18·14-s − 0.529·15-s − 0.0683·16-s + 0.242·17-s − 0.257·18-s − 1.81·19-s + 0.262·20-s − 2.17·21-s + 0.593·22-s − 1.14·23-s + 1.20·24-s + 0.200·25-s − 0.344·26-s − 0.709·27-s + 1.07·28-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 | \( 1 + 1.81T + 8T^{2} \) |
| 3 | \( 1 - 6.14T + 27T^{2} \) |
| 7 | \( 1 + 34.0T + 343T^{2} \) |
| 11 | \( 1 + 33.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 25.1T + 2.19e3T^{2} \) |
| 19 | \( 1 + 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 126.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 282.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 8.51T + 5.06e4T^{2} \) |
| 41 | \( 1 + 23.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 105.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 74.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 680.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 435.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 365.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 338.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 69.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 768.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 344.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 231.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 905.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 439.T + 9.12e5T^{2} \) |
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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
−13.30192272874504658189856392293, −12.52340546619497837581704407598, −10.45839310770802373439127382208, −9.729766530506504011246302363472, −8.612549225138939970989476342271, −7.997160413055771134556300611316, −6.37886809109170774726421589445, −4.13131234168550266052121163117, −2.84262556424100325770735285673, 0,
2.84262556424100325770735285673, 4.13131234168550266052121163117, 6.37886809109170774726421589445, 7.997160413055771134556300611316, 8.612549225138939970989476342271, 9.729766530506504011246302363472, 10.45839310770802373439127382208, 12.52340546619497837581704407598, 13.30192272874504658189856392293
