| L(s) = 1 | + (−4.21 − 3.77i)2-s − 3.25·3-s + (3.54 + 31.8i)4-s + (13.7 + 12.2i)6-s + 112. i·7-s + (105. − 147. i)8-s − 232.·9-s − 575. i·11-s + (−11.5 − 103. i)12-s − 117.·13-s + (425. − 475. i)14-s + (−998. + 225. i)16-s + 223. i·17-s + (979. + 876. i)18-s − 1.75e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.745 − 0.666i)2-s − 0.208·3-s + (0.110 + 0.993i)4-s + (0.155 + 0.139i)6-s + 0.869i·7-s + (0.580 − 0.814i)8-s − 0.956·9-s − 1.43i·11-s + (−0.0231 − 0.207i)12-s − 0.193·13-s + (0.579 − 0.647i)14-s + (−0.975 + 0.220i)16-s + 0.187i·17-s + (0.712 + 0.637i)18-s − 1.11i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9118378546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9118378546\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.21 + 3.77i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.25T + 243T^{2} \) |
| 7 | \( 1 - 112. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 575. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 117.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 223. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.75e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.36e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.86e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.73e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.15e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.10e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.41e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.20e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.12e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.23e4iT - 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30507401342935557142225972226, −10.90672378126139244378806919795, −9.363864479289640609107961430073, −8.789110292869134927292504514864, −7.86322324618228210208256486377, −6.38025341941588263743084377925, −5.25483512134080645828004726077, −3.41570624445068850040912604721, −2.46786687558901697843389407054, −0.73519621989485821730676743971,
0.57679958746200084105866219103, 2.17510129313125784240109617837, 4.26443467644214106775838983489, 5.47504121308507493231159105537, 6.63923366720695913145448024223, 7.53151425103356300359872652082, 8.460739409958444265136869600547, 9.715796306740024400680075511288, 10.34010201550638540833396470269, 11.38289626365029576875588422596
