L(s) = 1 | + (−442. + 442. i)3-s + (−2.49e3 − 2.49e3i)5-s − 3.93e4i·7-s − 2.13e5i·9-s + (3.11e5 + 3.11e5i)11-s + (−1.47e6 + 1.47e6i)13-s + 2.20e6·15-s + 6.43e6·17-s + (3.94e6 − 3.94e6i)19-s + (1.74e7 + 1.74e7i)21-s + 2.40e7i·23-s − 3.63e7i·25-s + (1.61e7 + 1.61e7i)27-s + (−1.10e8 + 1.10e8i)29-s − 2.26e8·31-s + ⋯ |
L(s) = 1 | + (−1.05 + 1.05i)3-s + (−0.357 − 0.357i)5-s − 0.885i·7-s − 1.20i·9-s + (0.582 + 0.582i)11-s + (−1.09 + 1.09i)13-s + 0.750·15-s + 1.10·17-s + (0.365 − 0.365i)19-s + (0.930 + 0.930i)21-s + 0.779i·23-s − 0.744i·25-s + (0.216 + 0.216i)27-s + (−0.996 + 0.996i)29-s − 1.42·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.3911941504\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3911941504\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (442. - 442. i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (2.49e3 + 2.49e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 + 3.93e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (-3.11e5 - 3.11e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (1.47e6 - 1.47e6i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 - 6.43e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (-3.94e6 + 3.94e6i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 - 2.40e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (1.10e8 - 1.10e8i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 + 2.26e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-2.38e8 - 2.38e8i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 - 1.45e9iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (1.31e8 + 1.31e8i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 + 2.03e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (1.22e9 + 1.22e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (2.81e9 + 2.81e9i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (-5.47e8 + 5.47e8i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (-1.17e10 + 1.17e10i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 + 1.29e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 2.25e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 3.10e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (3.09e10 - 3.09e10i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 + 7.63e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 5.73e10T + 7.15e21T^{2} \) |
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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
−11.13493510355947800785707659406, −9.868706073703301425415330186543, −9.472894112157311196839775697396, −7.67624847034528807234288780602, −6.67539592706836143272923121766, −5.21281035061636692490069173956, −4.50895434269088168140413235882, −3.59359223753092010903598145052, −1.47813630194145976235453255751, −0.14130292720202895971135870322,
0.76775300034272421876581133416, 2.09923172504382270891719696123, 3.44429096495672918928178784976, 5.40311334429219152993612742182, 5.86463370066432195914561426608, 7.18189887769764217901060354202, 7.88580529082327978772413134759, 9.362128792098197312375865967986, 10.70589445475215637836786749580, 11.66275240570088017160272952692