L(s) = 1 | + (0.707 + 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + 2.44i·6-s + (−6.80 + 1.64i)7-s − 2.82·8-s + (1.5 + 2.59i)9-s + (8.87 − 15.3i)11-s + (−2.99 + 1.73i)12-s + 2.00i·13-s + (−6.82 − 7.17i)14-s + (−2.00 − 3.46i)16-s + (−18.3 − 10.6i)17-s + (−2.12 + 3.67i)18-s + (−5.70 + 3.29i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.5 + 0.288i)3-s + (−0.249 + 0.433i)4-s + 0.408i·6-s + (−0.972 + 0.234i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.807 − 1.39i)11-s + (−0.249 + 0.144i)12-s + 0.154i·13-s + (−0.487 − 0.512i)14-s + (−0.125 − 0.216i)16-s + (−1.08 − 0.623i)17-s + (−0.117 + 0.204i)18-s + (−0.300 + 0.173i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.037741696\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.037741696\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.80 - 1.64i)T \) |
good | 11 | \( 1 + (-8.87 + 15.3i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 2.00iT - 169T^{2} \) |
| 17 | \( 1 + (18.3 + 10.6i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (5.70 - 3.29i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (19.7 + 34.1i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 25.1T + 841T^{2} \) |
| 31 | \( 1 + (-4.76 - 2.75i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (5.87 + 10.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 16.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 27.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (39.5 - 22.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-49.2 + 85.2i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-10.9 - 6.34i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-82.0 + 47.3i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-4.17 + 7.23i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 71.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-0.697 - 0.402i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (58.1 + 100. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 50.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-74.9 + 43.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 165. iT - 9.40e3T^{2} \) |
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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
−9.288198834847532631490558500901, −8.737376080398600473885933158027, −8.051178603430188573240342087299, −6.66573559282772576601204775039, −6.42542483460568924131406077851, −5.30172636291218815457929545282, −4.12646591381484179317445793693, −3.44317980131608036535598543208, −2.36697987803239533572608770056, −0.25181347492731309059846632278,
1.53307746954564189439928103000, 2.43975748692165006467841433801, 3.73444747231755251894263013620, 4.19763105103222008209402297913, 5.57107652676309572021276410194, 6.64009044621648993409052149772, 7.17051513661507644295312031008, 8.358240893674324806784026066685, 9.382427639280320256888181915579, 9.726915623456090471208734320412