| L(s) = 1 | + (0.366 − 1.36i)2-s + (−0.448 − 1.67i)3-s + (−1.73 − i)4-s + (−1.45 + 4.78i)5-s − 2.44·6-s + (−1.05 + 6.92i)7-s + (−2 + 1.99i)8-s + (−2.59 + 1.50i)9-s + (6.00 + 3.74i)10-s + (−6.50 + 11.2i)11-s + (−0.896 + 3.34i)12-s + (−0.863 + 0.863i)13-s + (9.06 + 3.96i)14-s + (8.65 + 0.293i)15-s + (1.99 + 3.46i)16-s + (−0.902 + 0.241i)17-s + ⋯ |
| L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.149 − 0.557i)3-s + (−0.433 − 0.250i)4-s + (−0.291 + 0.956i)5-s − 0.408·6-s + (−0.150 + 0.988i)7-s + (−0.250 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.600 + 0.374i)10-s + (−0.591 + 1.02i)11-s + (−0.0747 + 0.278i)12-s + (−0.0664 + 0.0664i)13-s + (0.647 + 0.283i)14-s + (0.577 + 0.0195i)15-s + (0.124 + 0.216i)16-s + (−0.0531 + 0.0142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.781204 + 0.474038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.781204 + 0.474038i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 3 | \( 1 + (0.448 + 1.67i)T \) |
| 5 | \( 1 + (1.45 - 4.78i)T \) |
| 7 | \( 1 + (1.05 - 6.92i)T \) |
| good | 11 | \( 1 + (6.50 - 11.2i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (0.863 - 0.863i)T - 169iT^{2} \) |
| 17 | \( 1 + (0.902 - 0.241i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (5.84 - 3.37i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-14.5 - 3.90i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 31.3iT - 841T^{2} \) |
| 31 | \( 1 + (-23.4 + 40.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (9.80 - 36.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 64.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-17.7 + 17.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (12.9 - 48.1i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-19.3 - 72.2i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-31.0 - 17.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (54.5 + 94.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-60.4 + 16.2i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 74.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-21.5 - 80.2i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-83.9 + 48.4i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-31.9 + 31.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (44.1 - 25.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (83.5 + 83.5i)T + 9.40e3iT^{2} \) |
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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
−12.24235533946003917778887378887, −11.49633239968153043998244071272, −10.53068574308168995037508838557, −9.587078307373438382920767358598, −8.315222824015919000390183926887, −7.18118349194200621754172888844, −6.10370883910866855039974249962, −4.81545401523669008837137480746, −3.11231256924189122392530957388, −2.07876536354279221135583665389,
0.48328304602846934425516382099, 3.48093690270757112833405729001, 4.58690471437317897323472815958, 5.49867747495015822115799693116, 6.81208131684245247828826958560, 8.068383307758744043605959477404, 8.773269181314593059810542052256, 9.970411450330412084873555347535, 10.93062553085781407970460299370, 12.05504779947169348894652518273
