L(s) = 1 | + (−1 − i)2-s + (2.14 + 2.09i)3-s + 2i·4-s + (1.13 − 4.86i)5-s + (−0.0523 − 4.24i)6-s + (2.31 − 6.60i)7-s + (2 − 2i)8-s + (0.222 + 8.99i)9-s + (−6.00 + 3.73i)10-s − 16.9i·11-s + (−4.18 + 4.29i)12-s + (−10.2 + 10.2i)13-s + (−8.91 + 4.29i)14-s + (12.6 − 8.07i)15-s − 4·16-s + (8.79 − 8.79i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (0.715 + 0.698i)3-s + 0.5i·4-s + (0.227 − 0.973i)5-s + (−0.00872 − 0.707i)6-s + (0.330 − 0.943i)7-s + (0.250 − 0.250i)8-s + (0.0246 + 0.999i)9-s + (−0.600 + 0.373i)10-s − 1.54i·11-s + (−0.349 + 0.357i)12-s + (−0.786 + 0.786i)13-s + (−0.637 + 0.306i)14-s + (0.842 − 0.538i)15-s − 0.250·16-s + (0.517 − 0.517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.32453 - 0.818758i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32453 - 0.818758i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (-2.14 - 2.09i)T \) |
| 5 | \( 1 + (-1.13 + 4.86i)T \) |
| 7 | \( 1 + (-2.31 + 6.60i)T \) |
good | 11 | \( 1 + 16.9iT - 121T^{2} \) |
| 13 | \( 1 + (10.2 - 10.2i)T - 169iT^{2} \) |
| 17 | \( 1 + (-8.79 + 8.79i)T - 289iT^{2} \) |
| 19 | \( 1 - 24.7T + 361T^{2} \) |
| 23 | \( 1 + (-19.2 + 19.2i)T - 529iT^{2} \) |
| 29 | \( 1 - 1.67T + 841T^{2} \) |
| 31 | \( 1 - 36.8iT - 961T^{2} \) |
| 37 | \( 1 + (-40.5 + 40.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 0.885T + 1.68e3T^{2} \) |
| 43 | \( 1 + (9.87 + 9.87i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (33.7 - 33.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (11.9 - 11.9i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 50.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (4.46 - 4.46i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 137. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-53.3 + 53.3i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (60.0 + 60.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 51.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (0.274 + 0.274i)T + 9.40e3iT^{2} \) |
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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
−11.73672750376133553344426738237, −10.87129876068116904502640049489, −9.854584916746385444373125930627, −9.104253892265899389816262646015, −8.293512799266217346303782573187, −7.29859274531306661606589600071, −5.27989059807420103031360446031, −4.23179364922959627040213647500, −2.94651247651673935087375856898, −1.05261901623184565059869763912,
1.85618793745476090557090404556, 3.03764645007367156350724478923, 5.20712936177019833465806910702, 6.43023465789370610598359869194, 7.49714397288020969138102087341, 7.950026608270931968217451550704, 9.549811342589750564720280557698, 9.811217289553898998461876201369, 11.42025195907949893172378949667, 12.34245301824387648666733763742