| L(s) = 1 | + (0.739 − 0.673i)2-s + (0.932 − 0.361i)3-s + (0.0922 − 0.995i)4-s + (−0.573 − 0.355i)5-s + (0.445 − 0.895i)6-s + (−0.276 + 0.365i)7-s + (−0.602 − 0.798i)8-s + (0.739 − 0.673i)9-s + (−0.663 + 0.124i)10-s + (1.64 − 1.49i)11-s + (−0.273 − 0.961i)12-s + (2.23 − 2.96i)13-s + (0.0422 + 0.456i)14-s + (−0.663 − 0.124i)15-s + (−0.982 − 0.183i)16-s + (−1.68 − 3.38i)17-s + ⋯ |
| L(s) = 1 | + (0.522 − 0.476i)2-s + (0.538 − 0.208i)3-s + (0.0461 − 0.497i)4-s + (−0.256 − 0.158i)5-s + (0.181 − 0.365i)6-s + (−0.104 + 0.138i)7-s + (−0.213 − 0.282i)8-s + (0.246 − 0.224i)9-s + (−0.209 + 0.0392i)10-s + (0.494 − 0.450i)11-s + (−0.0789 − 0.277i)12-s + (0.620 − 0.821i)13-s + (0.0113 + 0.121i)14-s + (−0.171 − 0.0320i)15-s + (−0.245 − 0.0459i)16-s + (−0.408 − 0.821i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0104 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0104 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58577 - 1.56934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58577 - 1.56934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.739 + 0.673i)T \) |
| 3 | \( 1 + (-0.932 + 0.361i)T \) |
| 103 | \( 1 + (-0.174 - 10.1i)T \) |
| good | 5 | \( 1 + (0.573 + 0.355i)T + (2.22 + 4.47i)T^{2} \) |
| 7 | \( 1 + (0.276 - 0.365i)T + (-1.91 - 6.73i)T^{2} \) |
| 11 | \( 1 + (-1.64 + 1.49i)T + (1.01 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.23 + 2.96i)T + (-3.55 - 12.5i)T^{2} \) |
| 17 | \( 1 + (1.68 + 3.38i)T + (-10.2 + 13.5i)T^{2} \) |
| 19 | \( 1 + (-3.27 - 1.26i)T + (14.0 + 12.8i)T^{2} \) |
| 23 | \( 1 + (0.565 + 0.515i)T + (2.12 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.58 - 0.979i)T + (12.9 + 25.9i)T^{2} \) |
| 31 | \( 1 + (-0.525 - 0.0982i)T + (28.9 + 11.1i)T^{2} \) |
| 37 | \( 1 + (-1.60 - 5.64i)T + (-31.4 + 19.4i)T^{2} \) |
| 41 | \( 1 + (5.69 - 3.52i)T + (18.2 - 36.7i)T^{2} \) |
| 43 | \( 1 + (0.944 - 3.32i)T + (-36.5 - 22.6i)T^{2} \) |
| 47 | \( 1 + 0.240T + 47T^{2} \) |
| 53 | \( 1 + (1.76 + 0.683i)T + (39.1 + 35.7i)T^{2} \) |
| 59 | \( 1 + (-2.46 - 3.26i)T + (-16.1 + 56.7i)T^{2} \) |
| 61 | \( 1 + (-5.54 - 11.1i)T + (-36.7 + 48.6i)T^{2} \) |
| 67 | \( 1 + (-1.35 - 1.78i)T + (-18.3 + 64.4i)T^{2} \) |
| 71 | \( 1 + (-0.534 + 0.330i)T + (31.6 - 63.5i)T^{2} \) |
| 73 | \( 1 + (-3.59 + 2.22i)T + (32.5 - 65.3i)T^{2} \) |
| 79 | \( 1 + (-1.63 - 1.00i)T + (35.2 + 70.7i)T^{2} \) |
| 83 | \( 1 + (-1.98 + 2.62i)T + (-22.7 - 79.8i)T^{2} \) |
| 89 | \( 1 + (0.227 + 2.45i)T + (-87.4 + 16.3i)T^{2} \) |
| 97 | \( 1 + (-4.17 + 8.38i)T + (-58.4 - 77.4i)T^{2} \) |
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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
−10.42901307831630054099521555499, −9.604551174176676071328580021655, −8.664338654566392134177788782663, −7.896634381778744711776906922049, −6.71750797018109368471244484583, −5.80140834484080249556432507689, −4.64581960587446162002023154981, −3.55248708789433091964068746476, −2.69802233583570464907881559464, −1.09056769581038956296770155088,
1.95220770987306464753637132770, 3.51796565946541630692476560915, 4.10574503920734727728669500176, 5.28029565582510823201720739383, 6.49391209964506491068447061467, 7.17319881511649978788942112019, 8.159755888489345426328759233452, 9.003546086037792940708838713268, 9.799056926479958583556203207666, 10.96495545652061187686364905465
