L(s) = 1 | + (1 − i)2-s + (−1.86 − 2.35i)3-s − 2i·4-s + (1.91 − 4.61i)5-s + (−4.21 − 0.488i)6-s + (6.26 − 3.12i)7-s + (−2 − 2i)8-s + (−2.05 + 8.76i)9-s + (−2.70 − 6.53i)10-s − 0.117i·11-s + (−4.70 + 3.72i)12-s + (−9.72 − 9.72i)13-s + (3.13 − 9.39i)14-s + (−14.4 + 4.09i)15-s − 4·16-s + (13.8 + 13.8i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (−0.620 − 0.783i)3-s − 0.5i·4-s + (0.383 − 0.923i)5-s + (−0.702 − 0.0814i)6-s + (0.894 − 0.446i)7-s + (−0.250 − 0.250i)8-s + (−0.228 + 0.973i)9-s + (−0.270 − 0.653i)10-s − 0.0106i·11-s + (−0.391 + 0.310i)12-s + (−0.747 − 0.747i)13-s + (0.223 − 0.670i)14-s + (−0.961 + 0.273i)15-s − 0.250·16-s + (0.815 + 0.815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.419016 - 1.61390i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.419016 - 1.61390i\) |
\(L(2)\) |
|
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 | \( 1 + (1.86 + 2.35i)T \) |
| 5 | \( 1 + (-1.91 + 4.61i)T \) |
| 7 | \( 1 + (-6.26 + 3.12i)T \) |
good | 11 | \( 1 + 0.117iT - 121T^{2} \) |
| 13 | \( 1 + (9.72 + 9.72i)T + 169iT^{2} \) |
| 17 | \( 1 + (-13.8 - 13.8i)T + 289iT^{2} \) |
| 19 | \( 1 + 29.7T + 361T^{2} \) |
| 23 | \( 1 + (2.25 + 2.25i)T + 529iT^{2} \) |
| 29 | \( 1 - 46.0T + 841T^{2} \) |
| 31 | \( 1 - 1.50iT - 961T^{2} \) |
| 37 | \( 1 + (-5.32 - 5.32i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 13.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36.8 + 36.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-29.7 - 29.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-59.8 - 59.8i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 84.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (34.4 + 34.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 77.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-41.3 - 41.3i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 0.865iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-99.0 + 99.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (15.7 - 15.7i)T - 9.40e3iT^{2} \) |
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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
−12.21106375195387321185407276234, −10.79136567687928690001968974321, −10.23816284431982839194977453745, −8.578261826932892363488212874568, −7.73023807749212907113431988487, −6.26849267134547143251367956592, −5.27443557403322426359563709147, −4.39139090420537416487442639210, −2.18049924974975951475462256203, −0.897010229355952556639695721454,
2.57709143269618267068702354630, 4.20797962246606298799174097360, 5.19538818377335946575214236323, 6.21736825555791917690033082025, 7.18755012112163832965533662965, 8.578393996319277690179501256210, 9.748855975611572126127277487056, 10.69451926682076469861449640478, 11.63771445639781537556629213037, 12.28974775999265749782017598598