L(s) = 1 | − 1.41i·2-s + (−1.86 + 2.35i)3-s − 2.00·4-s − 2.23i·5-s + (3.32 + 2.63i)6-s + 2.64·7-s + 2.82i·8-s + (−2.07 − 8.75i)9-s − 3.16·10-s + 19.0i·11-s + (3.72 − 4.70i)12-s − 1.55·13-s − 3.74i·14-s + (5.26 + 4.16i)15-s + 4.00·16-s + 28.9i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.620 + 0.784i)3-s − 0.500·4-s − 0.447i·5-s + (0.554 + 0.438i)6-s + 0.377·7-s + 0.353i·8-s + (−0.230 − 0.973i)9-s − 0.316·10-s + 1.73i·11-s + (0.310 − 0.392i)12-s − 0.119·13-s − 0.267i·14-s + (0.350 + 0.277i)15-s + 0.250·16-s + 1.70i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.932430 + 0.451326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.932430 + 0.451326i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (1.86 - 2.35i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
good | 11 | \( 1 - 19.0iT - 121T^{2} \) |
| 13 | \( 1 + 1.55T + 169T^{2} \) |
| 17 | \( 1 - 28.9iT - 289T^{2} \) |
| 19 | \( 1 - 23.3T + 361T^{2} \) |
| 23 | \( 1 - 23.7iT - 529T^{2} \) |
| 29 | \( 1 - 18.0iT - 841T^{2} \) |
| 31 | \( 1 - 7.88T + 961T^{2} \) |
| 37 | \( 1 + 20.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 50.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 31.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 15.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 78.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 89.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 56.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 28.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 120.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 72.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 104. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 36.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 57.1T + 9.40e3T^{2} \) |
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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
−12.20002194599688138013118960204, −11.33857707809562504347622628467, −10.25626898397338509001470487701, −9.700541094167091089486955533946, −8.631463198886955434365433577508, −7.23934333900540179666817605307, −5.59697580120872638807116874615, −4.71044333426339228365189124851, −3.68460624246411197508965494318, −1.61560320777198558239218585429,
0.67554090338942628777988336272, 2.95718588286098437254588925749, 4.94177250645462411399581550992, 5.88063907359864452955870269075, 6.85405758589820321582395274760, 7.76710624031926933324378496423, 8.693573369676617789734624688264, 10.09479263927927027243867331997, 11.36257313044560349935331776035, 11.76200142489651385365725450557