L(s) = 1 | − 4.37i·2-s + 12.8·4-s + 13.0·5-s − 213. i·7-s − 196. i·8-s − 57.3i·10-s − 174.·11-s − 394.·13-s − 933.·14-s − 448.·16-s + 1.40e3·17-s − 100. i·19-s + 168.·20-s + 761. i·22-s + (1.45e3 + 2.07e3i)23-s + ⋯ |
L(s) = 1 | − 0.773i·2-s + 0.401·4-s + 0.234·5-s − 1.64i·7-s − 1.08i·8-s − 0.181i·10-s − 0.433·11-s − 0.647·13-s − 1.27·14-s − 0.437·16-s + 1.17·17-s − 0.0637i·19-s + 0.0940·20-s + 0.335i·22-s + (0.574 + 0.818i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00355i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.804179389\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804179389\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (-1.45e3 - 2.07e3i)T \) |
good | 2 | \( 1 + 4.37iT - 32T^{2} \) |
| 5 | \( 1 - 13.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 213. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 174.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 394.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.40e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 100. iT - 2.47e6T^{2} \) |
| 29 | \( 1 + 5.00e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.98e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.93e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.11e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 6.97e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 5.35e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.19e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.69e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.40e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.38e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 2.16e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.03e5iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 8.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.21e4iT - 8.58e9T^{2} \) |
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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
−10.95518873132661040093208008890, −10.18508727476822769007027635787, −9.572664850056264909259720223732, −7.64203712086644867073848063127, −7.21956666530834598559277214763, −5.75045986361812950490002823294, −4.19032709038454123693024964881, −3.17299505984284235291340565407, −1.70591150651442213579514778265, −0.50158849520204737718563583703,
1.95236006040153994752223667466, 2.96694872354229079892913504371, 5.21147187560412135960109066833, 5.68313340023827874959563624865, 6.86229189127118002454143529854, 7.947645587716140375059559193987, 8.818835289169755457313794510548, 9.910292361709096700963503242449, 11.14778828433690346274722878462, 12.09978213159141917882193103839