L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s + (1 + 1.73i)5-s + (−2 + 3.46i)7-s − 3·8-s − 1.99·10-s + (−0.5 + 0.866i)11-s + (1 + 1.73i)13-s + (−1.99 − 3.46i)14-s + (0.500 − 0.866i)16-s − 2·17-s + (−0.999 + 1.73i)20-s + (−0.499 − 0.866i)22-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s − 1.99·26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s + (0.447 + 0.774i)5-s + (−0.755 + 1.30i)7-s − 1.06·8-s − 0.632·10-s + (−0.150 + 0.261i)11-s + (0.277 + 0.480i)13-s + (−0.534 − 0.925i)14-s + (0.125 − 0.216i)16-s − 0.485·17-s + (−0.223 + 0.387i)20-s + (−0.106 − 0.184i)22-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.173032 - 0.981313i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.173032 - 0.981313i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.38499618536219792262561892790, −9.623449107651876929411107624495, −8.754457252138944694619381754352, −8.203227840747362531681980236830, −6.96491256996299526289799693796, −6.35361440076113329356378593048, −5.90769682378417692738229776527, −4.35661074767739997125207210413, −2.84225982471357667634192984584, −2.44774566243421729489229678915,
0.51860870922335421073026234582, 1.57521238105964544316548082350, 3.01607619568294404647971627394, 4.05458031273644858220583848610, 5.34375729188104288795185261150, 6.14083880694584122465527035223, 7.04621176794054976283679933517, 8.114422174761065799486128789737, 9.153598643735392898116669754953, 9.796881391026020970327192691028