| L(s) = 1 | + 2.88·2-s − 15.5·3-s − 55.6·4-s − 133. i·5-s − 44.9·6-s + 59.3i·7-s − 345.·8-s + 243·9-s − 386. i·10-s + 578. i·11-s + 868.·12-s + 943.·13-s + 171. i·14-s + 2.08e3i·15-s + 2.56e3·16-s + 5.43e3i·17-s + ⋯ |
| L(s) = 1 | + 0.360·2-s − 0.577·3-s − 0.870·4-s − 1.07i·5-s − 0.208·6-s + 0.172i·7-s − 0.673·8-s + 0.333·9-s − 0.386i·10-s + 0.434i·11-s + 0.502·12-s + 0.429·13-s + 0.0623i·14-s + 0.618i·15-s + 0.627·16-s + 1.10i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.801425 + 0.544566i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801425 + 0.544566i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 15.5T \) |
| 23 | \( 1 + (4.48e3 - 1.13e4i)T \) |
| good | 2 | \( 1 - 2.88T + 64T^{2} \) |
| 5 | \( 1 + 133. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 59.3iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 578. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 943.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 5.43e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 3.90e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 - 1.12e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.51e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.09e3iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.34e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 6.63e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 2.58e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 6.09e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 7.68e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.86e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 2.45e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 5.11e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 1.04e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 6.99e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 2.88e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 9.05e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.69e6iT - 8.32e11T^{2} \) |
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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
−13.43377536079180433288553035708, −12.68423176377064563456161679592, −11.88087775355954710851596869678, −10.25092603038368743829991401348, −9.107573318220554129680810554743, −8.073278336619147535956979386570, −6.06113066910636318349616770606, −5.02345699041373744497562709994, −3.92375509262454276842593392376, −1.23229279154924785281430414574,
0.43591388171762773094685104144, 3.03803788090141403865772711203, 4.51578976144393837490594354788, 5.90214333683574288497791224866, 7.09847058973106074332401904967, 8.724918920310859424749783694111, 10.08148760864053369553392773164, 11.07439805545545725751268946915, 12.20534886791291291884052157860, 13.50892134595455875681686399781
