| L(s) = 1 | + 1.41·2-s + 2.00·4-s + (1.68 + 4.70i)5-s + 9.41i·7-s + 2.82·8-s + (2.37 + 6.65i)10-s + 2.35i·11-s − 1.53i·13-s + 13.3i·14-s + 4.00·16-s − 11.0·17-s + 7.09·19-s + (3.36 + 9.41i)20-s + 3.32i·22-s + 8.19·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + (0.336 + 0.941i)5-s + 1.34i·7-s + 0.353·8-s + (0.237 + 0.665i)10-s + 0.214i·11-s − 0.118i·13-s + 0.951i·14-s + 0.250·16-s − 0.652·17-s + 0.373·19-s + (0.168 + 0.470i)20-s + 0.151i·22-s + 0.356·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.706065093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.706065093\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.68 - 4.70i)T \) |
| good | 7 | \( 1 - 9.41iT - 49T^{2} \) |
| 11 | \( 1 - 2.35iT - 121T^{2} \) |
| 13 | \( 1 + 1.53iT - 169T^{2} \) |
| 17 | \( 1 + 11.0T + 289T^{2} \) |
| 19 | \( 1 - 7.09T + 361T^{2} \) |
| 23 | \( 1 - 8.19T + 529T^{2} \) |
| 29 | \( 1 - 17.9iT - 841T^{2} \) |
| 31 | \( 1 + 58.7T + 961T^{2} \) |
| 37 | \( 1 - 20.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 48.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 3.55iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 69.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 69.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 39.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 115.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 102. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 120. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 18.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 161.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 88.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 140. iT - 9.40e3T^{2} \) |
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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
−10.52098205475010706960093582125, −9.441780940299847166583172949769, −8.732623494256231822656014936932, −7.45654868198913620698313369465, −6.73558728837103679813286993100, −5.76890762284819234944958753012, −5.20740341088000801923529005678, −3.78627058362097608217463907586, −2.75963635184987756567972990259, −1.96614560408819148257123798214,
0.68960711474677156548175819951, 1.96531932501836660983375195941, 3.55090571928560232937513625784, 4.35852505856579069263400468637, 5.18154333723176557519721111748, 6.17432769499130133303055502572, 7.15302217053292390489731862330, 7.919976531392469386053576942761, 9.028242782107334119864434484076, 9.814560841563972487670184147770
