| L(s) = 1 | − 5.08·2-s + 2.28·3-s + 17.8·4-s + 18.7·5-s − 11.6·6-s − 50.0·8-s − 21.7·9-s − 95.1·10-s − 23.2·11-s + 40.8·12-s + 75.2·13-s + 42.8·15-s + 111.·16-s − 17·17-s + 110.·18-s − 123.·19-s + 334.·20-s + 118.·22-s + 54.4·23-s − 114.·24-s + 225.·25-s − 382.·26-s − 111.·27-s − 74.8·29-s − 217.·30-s − 252.·31-s − 167.·32-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.440·3-s + 2.23·4-s + 1.67·5-s − 0.791·6-s − 2.21·8-s − 0.806·9-s − 3.00·10-s − 0.636·11-s + 0.982·12-s + 1.60·13-s + 0.737·15-s + 1.74·16-s − 0.242·17-s + 1.44·18-s − 1.49·19-s + 3.73·20-s + 1.14·22-s + 0.493·23-s − 0.974·24-s + 1.80·25-s − 2.88·26-s − 0.795·27-s − 0.479·29-s − 1.32·30-s − 1.46·31-s − 0.927·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 5.08T + 8T^{2} \) |
| 3 | \( 1 - 2.28T + 27T^{2} \) |
| 5 | \( 1 - 18.7T + 125T^{2} \) |
| 11 | \( 1 + 23.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 75.2T + 2.19e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 54.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 74.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 43.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 6.84T + 1.03e5T^{2} \) |
| 53 | \( 1 + 302.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 296.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 523.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 543.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 606.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 219.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 795.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 770.T + 9.12e5T^{2} \) |
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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
−9.155641678341415453808461921879, −8.769936225057474617123803662179, −8.178597716644825153839895032124, −6.89677382046111634415718170811, −6.19209039654950439916328488336, −5.41420756548562288637953937369, −3.29131749989749724673653267536, −2.15611880950950487907120936254, −1.59404973258436100962838193317, 0,
1.59404973258436100962838193317, 2.15611880950950487907120936254, 3.29131749989749724673653267536, 5.41420756548562288637953937369, 6.19209039654950439916328488336, 6.89677382046111634415718170811, 8.178597716644825153839895032124, 8.769936225057474617123803662179, 9.155641678341415453808461921879
