| L(s) = 1 | + (−13.5 + 7.79i)3-s + (127. + 73.7i)5-s + (327. + 100. i)7-s + (121.5 − 210. i)9-s + (1.00e3 + 1.74e3i)11-s + 147. i·13-s − 2.29e3·15-s + (5.48e3 − 3.16e3i)17-s + (−589. − 340. i)19-s + (−5.21e3 + 1.19e3i)21-s + (9.31e3 − 1.61e4i)23-s + (3.05e3 + 5.28e3i)25-s + 3.78e3i·27-s − 3.10e4·29-s + (1.07e4 − 6.18e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.288i)3-s + (1.02 + 0.589i)5-s + (0.956 + 0.293i)7-s + (0.166 − 0.288i)9-s + (0.757 + 1.31i)11-s + 0.0671i·13-s − 0.680·15-s + (1.11 − 0.645i)17-s + (−0.0859 − 0.0496i)19-s + (−0.562 + 0.129i)21-s + (0.765 − 1.32i)23-s + (0.195 + 0.338i)25-s + 0.192i·27-s − 1.27·29-s + (0.359 − 0.207i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.559830323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559830323\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (13.5 - 7.79i)T \) |
| 7 | \( 1 + (-327. - 100. i)T \) |
| good | 5 | \( 1 + (-127. - 73.7i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-1.00e3 - 1.74e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 147. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-5.48e3 + 3.16e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (589. + 340. i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-9.31e3 + 1.61e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + 3.10e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.07e4 + 6.18e3i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (1.53e4 - 2.65e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 5.01e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.91e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.45e5 - 8.42e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-5.62e4 - 9.75e4i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (5.42e4 - 3.13e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.31e5 + 1.33e5i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-9.60e4 - 1.66e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.83e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-1.35e5 + 7.84e4i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.38e5 - 5.86e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 4.97e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (9.52e5 + 5.50e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + 3.65e5iT - 8.32e11T^{2} \) |
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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
−11.81887892713452129236934153267, −10.80552371987515262478750041456, −9.913095356745960353594158842151, −9.108931182750694403768598621297, −7.54225938725196403463004828558, −6.49543873721704339242588557019, −5.39782221126765721649356063986, −4.38111409658705126718148183759, −2.52290017165270201737110990063, −1.31274113086249934168628502740,
0.924657482895383728087860307298, 1.71074952979163119607721263807, 3.72424594073222676237035756907, 5.37257341980175994265502692890, 5.81183799112083386153440482829, 7.30365305509922410442770405905, 8.476286019814951310630600553947, 9.432054059931102604546115723663, 10.66785315004817002287911478409, 11.45450328467734122423408411446
