L(s) = 1 | + (0.5 − 0.866i)3-s + (−2.12 + 1.22i)5-s + (−0.499 − 0.866i)9-s + (2.12 + 1.22i)11-s − 4.89i·13-s + 2.44i·15-s + (2.12 + 1.22i)17-s + (−1 − 1.73i)19-s + (−6.36 + 3.67i)23-s + (0.499 − 0.866i)25-s − 0.999·27-s + 6·29-s + (4 − 6.92i)31-s + (2.12 − 1.22i)33-s + (−2 − 3.46i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.948 + 0.547i)5-s + (−0.166 − 0.288i)9-s + (0.639 + 0.369i)11-s − 1.35i·13-s + 0.632i·15-s + (0.514 + 0.297i)17-s + (−0.229 − 0.397i)19-s + (−1.32 + 0.766i)23-s + (0.0999 − 0.173i)25-s − 0.192·27-s + 1.11·29-s + (0.718 − 1.24i)31-s + (0.369 − 0.213i)33-s + (−0.328 − 0.569i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.263635353\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.263635353\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (2.12 - 1.22i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.12 - 1.22i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + (-2.12 - 1.22i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.36 - 3.67i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.34iT - 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.2iT - 71T^{2} \) |
| 73 | \( 1 + (12.7 + 7.34i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.48 + 4.89i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-10.6 + 6.12i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−8.439795498418020944569880840755, −7.976857214654895644121400582198, −7.42089339315576167436397592664, −6.54003281268467509015359001142, −5.84123497538938702547979711556, −4.67776462004007765434434209295, −3.67776826144763687796944812303, −3.10105826847871707999946141974, −1.89476967003019695923989206926, −0.46234514186710302445555291119,
1.18010167642330169905627542302, 2.55254286854752106108348586723, 3.77276181828754482905310660868, 4.21430116328806511180577838792, 4.96995876513802842666023947098, 6.13629041932152484475017100787, 6.87587312718681354021611826234, 7.85141131108840653049830250045, 8.590359211039601493606791777154, 8.906033190441031848491810721738