L(s) = 1 | + (0.654 + 0.755i)3-s + (−3.19 − 2.05i)5-s + (−0.0504 + 0.350i)7-s + (−0.142 + 0.989i)9-s + (0.990 + 0.636i)11-s + (2.04 + 4.47i)13-s + (−0.540 − 3.76i)15-s + (2.93 − 0.861i)17-s + (0.418 + 2.90i)19-s + (−0.298 + 0.191i)21-s + (0.101 + 0.117i)23-s + (3.92 + 8.58i)25-s + (−0.841 + 0.540i)27-s + 10.1·29-s + (−1.39 + 3.06i)31-s + ⋯ |
L(s) = 1 | + (0.378 + 0.436i)3-s + (−1.42 − 0.918i)5-s + (−0.0190 + 0.132i)7-s + (−0.0474 + 0.329i)9-s + (0.298 + 0.192i)11-s + (0.567 + 1.24i)13-s + (−0.139 − 0.971i)15-s + (0.711 − 0.208i)17-s + (0.0959 + 0.667i)19-s + (−0.0650 + 0.0418i)21-s + (0.0212 + 0.0244i)23-s + (0.784 + 1.71i)25-s + (−0.161 + 0.104i)27-s + 1.88·29-s + (−0.251 + 0.549i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21340 + 0.562040i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21340 + 0.562040i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (4.51 - 6.83i)T \) |
good | 5 | \( 1 + (3.19 + 2.05i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.0504 - 0.350i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.990 - 0.636i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.04 - 4.47i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.93 + 0.861i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.418 - 2.90i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (-0.101 - 0.117i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + (1.39 - 3.06i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 - 3.68T + 37T^{2} \) |
| 41 | \( 1 + (-3.40 + 0.998i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-3.71 + 1.09i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-2.23 - 2.57i)T + (-6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (8.63 + 2.53i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (3.22 - 7.06i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (8.93 - 5.74i)T + (25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (9.64 + 2.83i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (1.96 - 1.26i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.74 + 8.19i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-10.9 - 7.02i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (3.15 - 3.63i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 - 1.62T + 97T^{2} \) |
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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
−10.36238114082703544157302277863, −9.253416058622725375063063865962, −8.744206557957971842490402288935, −7.959733251617932056001440280261, −7.17288417194187752081931090399, −5.90242976475543670593487198125, −4.57304946252065192762788598695, −4.18757548885565471697122375431, −3.10212727144345123437372519747, −1.26861571601196649921896568827,
0.76889618699481320807474512988, 2.83229192839768700805777655760, 3.43194870097443108992795185160, 4.48363199208049931247774868184, 5.98161657389122402112477207566, 6.82286619792546144528181572348, 7.80778616720260986511516597441, 8.039085718284299890543926047995, 9.151546062324642186507158408787, 10.41027482634713840361163459890