| L(s) = 1 | + (2 + 3.46i)2-s + (−15.1 + 3.60i)3-s + (−7.99 + 13.8i)4-s + (23.5 − 40.8i)5-s + (−42.8 − 45.3i)6-s + (19.3 + 33.5i)7-s − 63.9·8-s + (216. − 109. i)9-s + 188.·10-s + (−60.5 − 104. i)11-s + (71.3 − 238. i)12-s + (−285. + 494. i)13-s + (−77.5 + 134. i)14-s + (−210. + 704. i)15-s + (−128 − 221. i)16-s + 303.·17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.972 + 0.231i)3-s + (−0.249 + 0.433i)4-s + (0.421 − 0.730i)5-s + (−0.485 − 0.513i)6-s + (0.149 + 0.259i)7-s − 0.353·8-s + (0.892 − 0.450i)9-s + 0.596·10-s + (−0.150 − 0.261i)11-s + (0.143 − 0.479i)12-s + (−0.468 + 0.811i)13-s + (−0.105 + 0.183i)14-s + (−0.241 + 0.808i)15-s + (−0.125 − 0.216i)16-s + 0.254·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6700411791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6700411791\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 - 3.46i)T \) |
| 3 | \( 1 + (15.1 - 3.60i)T \) |
| 11 | \( 1 + (60.5 + 104. i)T \) |
| good | 5 | \( 1 + (-23.5 + 40.8i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-19.3 - 33.5i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 13 | \( 1 + (285. - 494. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 303.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.18e3 - 3.78e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (3.82e3 + 6.62e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.51e3 - 2.62e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 6.00e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (9.82e3 - 1.70e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (5.32e3 + 9.22e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.00e4 + 1.74e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (3.32e3 - 5.75e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-381. - 661. i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (3.44e4 - 5.96e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 8.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-2.92e4 - 5.06e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.73e4 - 3.00e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 7.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.12e4 - 5.41e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−11.93268501756046400997428824465, −11.56009555890158791274980025276, −9.912320528757359736965863481651, −9.288240268520681048918699306171, −7.88658747362848664851829004830, −6.77229513979879080472606683169, −5.53663662809099147415136206014, −5.11240054671539407346153529031, −3.73805454358713982642467868947, −1.51979944973851244610110796911,
0.20708502338649190215492948714, 1.71178498442096096296723610541, 3.13630368336940401374123493733, 4.72336425269515453253697162471, 5.66370402568997719764009740413, 6.74762066845959366630507057881, 7.78851576632789761740869762243, 9.598287315790257245710455194595, 10.46557881213100964638872694515, 10.94602518632132278531904491929
