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
−8.906033190441031848491810721738, −8.590359211039601493606791777154, −7.85141131108840653049830250045, −6.87587312718681354021611826234, −6.13629041932152484475017100787, −4.96995876513802842666023947098, −4.21430116328806511180577838792, −3.77276181828754482905310660868, −2.55254286854752106108348586723, −1.18010167642330169905627542302,
0.46234514186710302445555291119, 1.89476967003019695923989206926, 3.10105826847871707999946141974, 3.67776826144763687796944812303, 4.67776462004007765434434209295, 5.84123497538938702547979711556, 6.54003281268467509015359001142, 7.42089339315576167436397592664, 7.976857214654895644121400582198, 8.439795498418020944569880840755