L(s) = 1 | + (1.22 − 1.22i)3-s + (1.69 − 1.69i)5-s + 5.74·7-s − 2.99i·9-s + (5.59 + 5.59i)11-s + (−13.5 − 13.5i)13-s − 4.16i·15-s + 19.7·17-s + (21.6 − 21.6i)19-s + (7.03 − 7.03i)21-s − 24.9·23-s + 19.2i·25-s + (−3.67 − 3.67i)27-s + (1.50 + 1.50i)29-s + 2.20i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.339 − 0.339i)5-s + 0.820·7-s − 0.333i·9-s + (0.508 + 0.508i)11-s + (−1.04 − 1.04i)13-s − 0.277i·15-s + 1.15·17-s + (1.14 − 1.14i)19-s + (0.334 − 0.334i)21-s − 1.08·23-s + 0.768i·25-s + (−0.136 − 0.136i)27-s + (0.0519 + 0.0519i)29-s + 0.0709i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.80379 - 0.718034i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.80379 - 0.718034i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-1.69 + 1.69i)T - 25iT^{2} \) |
| 7 | \( 1 - 5.74T + 49T^{2} \) |
| 11 | \( 1 + (-5.59 - 5.59i)T + 121iT^{2} \) |
| 13 | \( 1 + (13.5 + 13.5i)T + 169iT^{2} \) |
| 17 | \( 1 - 19.7T + 289T^{2} \) |
| 19 | \( 1 + (-21.6 + 21.6i)T - 361iT^{2} \) |
| 23 | \( 1 + 24.9T + 529T^{2} \) |
| 29 | \( 1 + (-1.50 - 1.50i)T + 841iT^{2} \) |
| 31 | \( 1 - 2.20iT - 961T^{2} \) |
| 37 | \( 1 + (-27.6 + 27.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (21.4 + 21.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 76.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (56.5 - 56.5i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-48.0 - 48.0i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (51.5 + 51.5i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (63.4 - 63.4i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 43.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 73.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 4.12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (38.4 - 38.4i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 52.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 23.1T + 9.40e3T^{2} \) |
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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
−12.27116236603601988649892398951, −11.40173716645377634829362583464, −9.986163166581230435605187024045, −9.274522895155907821618810733015, −7.922291582803701823095405703366, −7.36186560787187861588520572543, −5.73370262620505840261307968914, −4.69130062854786280716629153335, −2.91491663454029029810670624769, −1.32521575565122603045165404621,
1.86971880102786645922081865275, 3.49843320985271908252045059686, 4.81680572321576901431856270694, 6.05557824108513174711899916132, 7.47941956872004469345467952026, 8.361574790583027789890575129442, 9.649418644634103621436688347455, 10.19606153174793238023737012859, 11.60379136869704027620594655456, 12.12102298637039066591011473233