| L(s) = 1 | + (−21 − 16.9i)3-s + 169. i·5-s + 2·7-s + (153. + 712. i)9-s + 33.9i·11-s + 2.95e3·13-s + (2.88e3 − 3.56e3i)15-s + 4.48e3i·17-s − 5.25e3·19-s + (−42 − 33.9i)21-s − 1.02e4i·23-s − 1.31e4·25-s + (8.88e3 − 1.75e4i)27-s − 2.20e3i·29-s + 2.28e4·31-s + ⋯ |
| L(s) = 1 | + (−0.777 − 0.628i)3-s + 1.35i·5-s + 0.00583·7-s + (0.209 + 0.977i)9-s + 0.0255i·11-s + 1.34·13-s + (0.853 − 1.05i)15-s + 0.911i·17-s − 0.766·19-s + (−0.00453 − 0.00366i)21-s − 0.842i·23-s − 0.843·25-s + (0.451 − 0.892i)27-s − 0.0904i·29-s + 0.768·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8183188964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8183188964\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (21 + 16.9i)T \) |
| good | 5 | \( 1 - 169. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 2T + 1.17e5T^{2} \) |
| 11 | \( 1 - 33.9iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.95e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.48e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 5.25e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.02e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.20e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.28e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.40e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.67e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.40e3T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.79e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.92e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.26e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 6.25e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.38e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 6.82e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 7.30e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.40e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 4.96e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 3.86e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 2.81e5T + 8.32e11T^{2} \) |
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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
−11.70024162155993725173622569610, −10.69766134325806103040588097787, −10.48205384685107958544292833803, −8.640989358195149584938553670282, −7.55669228165606360592259965474, −6.43405612232792369341128446141, −6.04205749649615516870674866829, −4.27385367698987898980544536043, −2.80357064097396915466721767734, −1.42103927080781707987557928088,
0.27190927060308537420611369931, 1.38090979098309881164155549510, 3.61836347489956775401733642947, 4.71785903879708832661512800125, 5.53248234550219966178354264792, 6.65928994522426545702929126005, 8.317421652675071035950448798649, 9.081933227418080509057283708698, 10.04181287163406750227337818160, 11.19262266883399505293752683071
