L(s) = 1 | + (1.36 − 0.366i)2-s + (−1.67 − 0.448i)3-s + (1.73 − i)4-s + (−4.19 + 2.72i)5-s − 2.44·6-s + (−1.84 + 6.75i)7-s + (1.99 − 2i)8-s + (2.59 + 1.50i)9-s + (−4.73 + 5.25i)10-s + (7.12 + 12.3i)11-s + (−3.34 + 0.896i)12-s + (−2.75 + 2.75i)13-s + (−0.0509 + 9.89i)14-s + (8.23 − 2.67i)15-s + (1.99 − 3.46i)16-s + (−6.41 + 23.9i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.557 − 0.149i)3-s + (0.433 − 0.250i)4-s + (−0.839 + 0.544i)5-s − 0.408·6-s + (−0.263 + 0.964i)7-s + (0.249 − 0.250i)8-s + (0.288 + 0.166i)9-s + (−0.473 + 0.525i)10-s + (0.647 + 1.12i)11-s + (−0.278 + 0.0747i)12-s + (−0.212 + 0.212i)13-s + (−0.00363 + 0.707i)14-s + (0.549 − 0.178i)15-s + (0.124 − 0.216i)16-s + (−0.377 + 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.00884 + 0.892733i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00884 + 0.892733i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.366i)T \) |
| 3 | \( 1 + (1.67 + 0.448i)T \) |
| 5 | \( 1 + (4.19 - 2.72i)T \) |
| 7 | \( 1 + (1.84 - 6.75i)T \) |
good | 11 | \( 1 + (-7.12 - 12.3i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (2.75 - 2.75i)T - 169iT^{2} \) |
| 17 | \( 1 + (6.41 - 23.9i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-0.277 - 0.159i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (3.64 + 13.6i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 24.2iT - 841T^{2} \) |
| 31 | \( 1 + (7.62 + 13.1i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (2.28 - 0.611i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 29.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (11.7 - 11.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (81.7 - 21.9i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-72.6 - 19.4i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-31.7 + 18.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-54.2 + 93.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-17.6 + 66.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 20.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-56.5 - 15.1i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-44.5 - 25.7i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (82.8 - 82.8i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-95.6 - 55.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-68.2 - 68.2i)T + 9.40e3iT^{2} \) |
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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
−12.39342598908750884468833852666, −11.60025013888906814081454528729, −10.75237665290453008919787372108, −9.630771262203368845380559911526, −8.234140673657237023003841391943, −6.91885847479919911956332433975, −6.23651636801542504114227012058, −4.83356360587721801753291458462, −3.73346799250853268303930439201, −2.11127566577221693337121489452,
0.65554813988911445324829791987, 3.40603326239116368241919732792, 4.35087056385769628919362390950, 5.44737822216614751765648576482, 6.74469700321268135039431883510, 7.60287235328899683939233826370, 8.864735972431658006401867893439, 10.15614710733396553190921430746, 11.54429954087610144804607778681, 11.57035132178237894929035346852