| L(s) = 1 | + (−0.553 + 1.07i)2-s + (1.90 + 4.83i)3-s + (3.79 + 5.32i)4-s + (3.34 − 13.8i)5-s + (−6.24 − 0.627i)6-s + (4.72 + 1.63i)7-s + (−17.3 + 2.49i)8-s + (−19.7 + 18.4i)9-s + (12.9 + 11.2i)10-s + (−0.374 + 7.86i)11-s + (−18.5 + 28.5i)12-s + (26.1 + 75.4i)13-s + (−4.37 + 4.16i)14-s + (73.1 − 10.1i)15-s + (−10.1 + 29.4i)16-s + (1.14 + 2.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.195 + 0.379i)2-s + (0.367 + 0.930i)3-s + (0.474 + 0.666i)4-s + (0.299 − 1.23i)5-s + (−0.424 − 0.0426i)6-s + (0.255 + 0.0883i)7-s + (−0.768 + 0.110i)8-s + (−0.730 + 0.682i)9-s + (0.409 + 0.355i)10-s + (−0.0102 + 0.215i)11-s + (−0.445 + 0.685i)12-s + (0.556 + 1.60i)13-s + (−0.0834 + 0.0795i)14-s + (1.25 − 0.174i)15-s + (−0.159 + 0.459i)16-s + (0.0163 + 0.0357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.829494 + 1.72468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.829494 + 1.72468i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.90 - 4.83i)T \) |
| 23 | \( 1 + (85.6 - 69.5i)T \) |
| good | 2 | \( 1 + (0.553 - 1.07i)T + (-4.64 - 6.51i)T^{2} \) |
| 5 | \( 1 + (-3.34 + 13.8i)T + (-111. - 57.2i)T^{2} \) |
| 7 | \( 1 + (-4.72 - 1.63i)T + (269. + 212. i)T^{2} \) |
| 11 | \( 1 + (0.374 - 7.86i)T + (-1.32e3 - 126. i)T^{2} \) |
| 13 | \( 1 + (-26.1 - 75.4i)T + (-1.72e3 + 1.35e3i)T^{2} \) |
| 17 | \( 1 + (-1.14 - 2.50i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-29.7 - 13.5i)T + (4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-86.9 - 61.9i)T + (7.97e3 + 2.30e4i)T^{2} \) |
| 31 | \( 1 + (-14.9 - 5.97i)T + (2.15e4 + 2.05e4i)T^{2} \) |
| 37 | \( 1 + (4.93 - 16.7i)T + (-4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (159. + 38.6i)T + (6.12e4 + 3.15e4i)T^{2} \) |
| 43 | \( 1 + (2.57 + 6.44i)T + (-5.75e4 + 5.48e4i)T^{2} \) |
| 47 | \( 1 + (-547. + 315. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-228. - 263. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-332. + 114. i)T + (1.61e5 - 1.26e5i)T^{2} \) |
| 61 | \( 1 + (-278. - 353. i)T + (-5.35e4 + 2.20e5i)T^{2} \) |
| 67 | \( 1 + (696. - 33.1i)T + (2.99e5 - 2.85e4i)T^{2} \) |
| 71 | \( 1 + (322. + 501. i)T + (-1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-243. + 534. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (36.4 - 189. i)T + (-4.57e5 - 1.83e5i)T^{2} \) |
| 83 | \( 1 + (-15.1 - 62.5i)T + (-5.08e5 + 2.62e5i)T^{2} \) |
| 89 | \( 1 + (-194. + 1.35e3i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-910. + 955. i)T + (-4.34e4 - 9.11e5i)T^{2} \) |
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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
−12.07380979600497689386588426577, −11.50759226236857604763385244641, −10.10979539800676311139981843102, −8.891951747873454040885978191014, −8.720401601397025619973485802328, −7.42891241915852472555289571833, −5.97920581143568923624540095186, −4.75264887083822234974757129307, −3.69992287429203002970776560001, −1.94854596080051818512560166703,
0.832158257825284609004246358471, 2.36329852257325757291057175146, 3.19817863332324106849923230762, 5.71128391487334423212705278814, 6.42126682263261796471110996899, 7.44329737601633543510873562620, 8.496594369126875306795747080989, 9.929274908321287903342324789100, 10.65999285021892006186808560522, 11.44048765346303567345979713235
