| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 4i·5-s + 2.82i·8-s − 5.65·10-s + 2.82i·11-s + 12.7·13-s + 4.00·16-s − 4i·17-s + 22.6·19-s + 8.00i·20-s + 4.00·22-s − 36.7i·23-s + 9·25-s − 18i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.800i·5-s + 0.353i·8-s − 0.565·10-s + 0.257i·11-s + 0.979·13-s + 0.250·16-s − 0.235i·17-s + 1.19·19-s + 0.400i·20-s + 0.181·22-s − 1.59i·23-s + 0.359·25-s − 0.692i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.561681199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561681199\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4iT - 25T^{2} \) |
| 11 | \( 1 - 2.82iT - 121T^{2} \) |
| 13 | \( 1 - 12.7T + 169T^{2} \) |
| 17 | \( 1 + 4iT - 289T^{2} \) |
| 19 | \( 1 - 22.6T + 361T^{2} \) |
| 23 | \( 1 + 36.7iT - 529T^{2} \) |
| 29 | \( 1 + 32.5iT - 841T^{2} \) |
| 31 | \( 1 + 50.9T + 961T^{2} \) |
| 37 | \( 1 + 32T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 20T + 1.84e3T^{2} \) |
| 47 | \( 1 - 20iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 94.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 83.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 48T + 4.48e3T^{2} \) |
| 71 | \( 1 + 76.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 148T + 6.24e3T^{2} \) |
| 83 | \( 1 + 80iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 106iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 154.T + 9.40e3T^{2} \) |
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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
−9.610307458970324435192446740254, −8.848653056588670877399564576852, −8.215997516169116827841133213764, −7.10662046373853454793794181968, −5.91528427975366815224190816276, −4.99627272396817247424165704985, −4.13768526888750765063140877640, −3.05492151474207532749374526011, −1.71212627479738322386044140685, −0.56193555964805536961257468428,
1.38194832148221380729769100428, 3.15020572701284162199747998256, 3.84254587389051184293736457857, 5.35415218728470933272832447011, 5.87345101926073778281718802119, 7.10278913808860247598617952721, 7.38831352606443878770958447374, 8.638354291851642065026673946212, 9.221583154837629611110269202067, 10.30223366431066712669652741204
