L(s) = 1 | + (−4.78 + 3.01i)2-s − 25.4i·3-s + (13.7 − 28.8i)4-s − 25i·5-s + (76.7 + 121. i)6-s − 56.4·7-s + (21.0 + 179. i)8-s − 403.·9-s + (75.4 + 119. i)10-s + 261. i·11-s + (−734. − 350. i)12-s − 720. i·13-s + (270. − 170. i)14-s − 635.·15-s + (−643. − 796. i)16-s − 1.87e3·17-s + ⋯ |
L(s) = 1 | + (−0.845 + 0.533i)2-s − 1.63i·3-s + (0.431 − 0.902i)4-s − 0.447i·5-s + (0.870 + 1.38i)6-s − 0.435·7-s + (0.116 + 0.993i)8-s − 1.66·9-s + (0.238 + 0.378i)10-s + 0.650i·11-s + (−1.47 − 0.703i)12-s − 1.18i·13-s + (0.368 − 0.232i)14-s − 0.729·15-s + (−0.628 − 0.778i)16-s − 1.57·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0318210 - 0.545135i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0318210 - 0.545135i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (4.78 - 3.01i)T \) |
| 5 | \( 1 + 25iT \) |
good | 3 | \( 1 + 25.4iT - 243T^{2} \) |
| 7 | \( 1 + 56.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 261. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 720. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.87e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.99e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.70e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.49e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.81e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.14e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.60e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.68e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.45e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.24e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 8.18e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.16e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.92e4T + 8.58e9T^{2} \) |
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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
−14.63943489922002867465590135111, −13.20677671278805413310611686322, −12.42762060537959105946118417049, −10.85882712534808644129822373702, −9.168248492238149391517204494804, −7.929472615518512249182135200768, −6.97398241242138959002025300517, −5.71396772483353115874274391965, −2.00629340232989276197903019190, −0.38345842016112602331330427621,
2.93664359031754054838196958755, 4.38021881935717754243387996048, 6.73419710463704987266415017807, 8.899016883925746138676638612062, 9.451981900246634135585584144016, 10.91017559716701977948182259737, 11.28257189450929486535582814470, 13.30873577810901352629493080512, 14.94448089229218011456676125215, 15.98169956113949797730976656021