| L(s) = 1 | + i·2-s − 4-s − 2.74i·5-s + (−2.24 + 1.39i)7-s − i·8-s + 2.74·10-s + 1.52·11-s − 5.33·13-s + (−1.39 − 2.24i)14-s + 16-s − 2.78i·17-s + (4.01 + 1.68i)19-s + 2.74i·20-s + 1.52i·22-s − 6.49·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.22i·5-s + (−0.849 + 0.527i)7-s − 0.353i·8-s + 0.867·10-s + 0.461·11-s − 1.48·13-s + (−0.372 − 0.600i)14-s + 0.250·16-s − 0.676i·17-s + (0.921 + 0.387i)19-s + 0.613i·20-s + 0.326i·22-s − 1.35·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9892096358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9892096358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.24 - 1.39i)T \) |
| 19 | \( 1 + (-4.01 - 1.68i)T \) |
| good | 5 | \( 1 + 2.74iT - 5T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 + 5.33T + 13T^{2} \) |
| 17 | \( 1 + 2.78iT - 17T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 6.88iT - 29T^{2} \) |
| 31 | \( 1 - 0.830T + 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 0.668T + 43T^{2} \) |
| 47 | \( 1 + 4.83iT - 47T^{2} \) |
| 53 | \( 1 - 2.02iT - 53T^{2} \) |
| 59 | \( 1 - 5.53T + 59T^{2} \) |
| 61 | \( 1 - 7.03iT - 61T^{2} \) |
| 67 | \( 1 + 3.43iT - 67T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 - 12.5iT - 73T^{2} \) |
| 79 | \( 1 - 4.32iT - 79T^{2} \) |
| 83 | \( 1 - 14.4iT - 83T^{2} \) |
| 89 | \( 1 - 6.14T + 89T^{2} \) |
| 97 | \( 1 - 5.31T + 97T^{2} \) |
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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
−9.195493831234442323016826757090, −8.435043832132403651008810414538, −7.62701408412686921528204011850, −6.90144700316919734132803408213, −5.99809237112089142069108613237, −5.23952240505214312541982359769, −4.69105914103906218040366276630, −3.63492719686211668918315462436, −2.48038159672207276744913820784, −0.977530807949581123951408510949,
0.40547963138297813420673561325, 2.14747877858586417954991478588, 2.84894020079474726380984211482, 3.74733852983460621915575622512, 4.41600058187320193434019904813, 5.75751541429271664650487186821, 6.41038047342274014794826324616, 7.37918061221032753247485038496, 7.71080248822732140397110589724, 9.098393723548302634538748651854
