| L(s) = 1 | − 9.57·2-s + 7.84·3-s + 59.7·4-s − 75.1·6-s − 195.·7-s − 265.·8-s − 181.·9-s + 72.8·11-s + 468.·12-s − 301.·13-s + 1.87e3·14-s + 632.·16-s + 1.20e3·17-s + 1.73e3·18-s − 2.35e3·19-s − 1.53e3·21-s − 697.·22-s − 516.·23-s − 2.08e3·24-s + 2.88e3·26-s − 3.32e3·27-s − 1.16e4·28-s + 1.53e3·29-s + 1.12e3·31-s + 2.44e3·32-s + 571.·33-s − 1.15e4·34-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.503·3-s + 1.86·4-s − 0.851·6-s − 1.50·7-s − 1.46·8-s − 0.746·9-s + 0.181·11-s + 0.939·12-s − 0.495·13-s + 2.55·14-s + 0.617·16-s + 1.01·17-s + 1.26·18-s − 1.49·19-s − 0.759·21-s − 0.307·22-s − 0.203·23-s − 0.738·24-s + 0.838·26-s − 0.878·27-s − 2.81·28-s + 0.338·29-s + 0.210·31-s + 0.421·32-s + 0.0913·33-s − 1.71·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 2 | \( 1 + 9.57T + 32T^{2} \) |
| 3 | \( 1 - 7.84T + 243T^{2} \) |
| 7 | \( 1 + 195.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 72.8T + 1.61e5T^{2} \) |
| 13 | \( 1 + 301.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 516.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.53e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.12e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.97e4T + 1.15e8T^{2} \) |
| 47 | \( 1 - 1.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.35e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.51e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.11e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.57e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.79e4T + 8.58e9T^{2} \) |
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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
−8.859036195508357540233247824633, −8.131330866280327699858873776090, −7.36296903984809427608804942828, −6.47074049228840620682142714100, −5.81856346487294177909154483487, −4.04929434150445132674427458631, −2.87519674884014151795979484024, −2.30907198323852570545249833146, −0.839841241893991328725783339269, 0,
0.839841241893991328725783339269, 2.30907198323852570545249833146, 2.87519674884014151795979484024, 4.04929434150445132674427458631, 5.81856346487294177909154483487, 6.47074049228840620682142714100, 7.36296903984809427608804942828, 8.131330866280327699858873776090, 8.859036195508357540233247824633
