| L(s) = 1 | − 8.34·2-s − 9·3-s + 37.6·4-s − 107.·5-s + 75.1·6-s − 26.6·7-s − 47.1·8-s + 81·9-s + 896.·10-s + 121·11-s − 338.·12-s + 904.·13-s + 222.·14-s + 967.·15-s − 811.·16-s − 495.·17-s − 676.·18-s − 1.50e3·19-s − 4.04e3·20-s + 240.·21-s − 1.00e3·22-s + 2.39e3·23-s + 424.·24-s + 8.42e3·25-s − 7.55e3·26-s − 729·27-s − 1.00e3·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 0.577·3-s + 1.17·4-s − 1.92·5-s + 0.851·6-s − 0.205·7-s − 0.260·8-s + 0.333·9-s + 2.83·10-s + 0.301·11-s − 0.679·12-s + 1.48·13-s + 0.303·14-s + 1.10·15-s − 0.792·16-s − 0.415·17-s − 0.491·18-s − 0.954·19-s − 2.26·20-s + 0.118·21-s − 0.444·22-s + 0.943·23-s + 0.150·24-s + 2.69·25-s − 2.19·26-s − 0.192·27-s − 0.242·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3484711604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3484711604\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 8.34T + 32T^{2} \) |
| 5 | \( 1 + 107.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 26.6T + 1.68e4T^{2} \) |
| 13 | \( 1 - 904.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 495.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 410.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.85e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 788.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5T + 8.58e9T^{2} \) |
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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
−16.07809785890652776194182721632, −15.17492946747649701884565699229, −12.84285927985097752328433191834, −11.28812293790970773653276316684, −10.94095646036259825092598392044, −9.008339541023446007286035294706, −7.987752016260584808050827269045, −6.76955182103526619177115805629, −4.06079175080597074273269044081, −0.70022603535538289271875179639,
0.70022603535538289271875179639, 4.06079175080597074273269044081, 6.76955182103526619177115805629, 7.987752016260584808050827269045, 9.008339541023446007286035294706, 10.94095646036259825092598392044, 11.28812293790970773653276316684, 12.84285927985097752328433191834, 15.17492946747649701884565699229, 16.07809785890652776194182721632
