| L(s) = 1 | + 0.456i·2-s + 1.79·4-s + (0.456 + 2.18i)5-s + i·7-s + 1.73i·8-s + (−0.999 + 0.208i)10-s − 1.73·11-s + 0.208i·13-s − 0.456·14-s + 2.79·16-s + 3.00i·17-s + 2.20·19-s + (0.818 + 3.92i)20-s − 0.791i·22-s + 3.10i·23-s + ⋯ |
| L(s) = 1 | + 0.323i·2-s + 0.895·4-s + (0.204 + 0.978i)5-s + 0.377i·7-s + 0.612i·8-s + (−0.316 + 0.0660i)10-s − 0.522·11-s + 0.0578i·13-s − 0.122·14-s + 0.697·16-s + 0.729i·17-s + 0.506·19-s + (0.182 + 0.876i)20-s − 0.168i·22-s + 0.646i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20492 + 1.48237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20492 + 1.48237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.456 - 2.18i)T \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 - 0.456iT - 2T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 0.208iT - 13T^{2} \) |
| 17 | \( 1 - 3.00iT - 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 - 3.10iT - 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 0.582T + 31T^{2} \) |
| 37 | \( 1 + 8.16iT - 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 1.73iT - 47T^{2} \) |
| 53 | \( 1 + 1.37iT - 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 0.208T + 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 4.58iT - 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 + 8.66T + 89T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−10.37876496300197038281522315175, −9.582931155121847458944338677565, −8.435020500840356187283413593650, −7.49051744044333367648982493934, −7.03208752360052389409158717116, −5.90041839748926584625368956789, −5.53715204006036105135016651729, −3.81625389619872700209845439160, −2.77740693530013879583669126319, −1.89713942362452134568231293296,
0.878488367702562355798360889511, 2.13676905240918307572456650368, 3.26175817970916382456223331011, 4.49283027807220937115158583497, 5.42942591102412229167848976217, 6.35711525631442395683557098426, 7.38556193354714628015959539176, 8.024573625067698947294585721468, 9.127103790621161979293477164325, 9.888784255444233379242062813190
