L(s) = 1 | + (0.366 + 1.36i)2-s + (0.448 − 1.67i)3-s + (−1.73 + i)4-s + (−4.09 + 2.86i)5-s + 2.44·6-s + (−3.46 − 6.08i)7-s + (−2 − 1.99i)8-s + (−2.59 − 1.50i)9-s + (−5.41 − 4.54i)10-s + (−5.62 − 9.74i)11-s + (0.896 + 3.34i)12-s + (−5.84 − 5.84i)13-s + (7.04 − 6.95i)14-s + (2.96 + 8.13i)15-s + (1.99 − 3.46i)16-s + (24.8 + 6.65i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (0.149 − 0.557i)3-s + (−0.433 + 0.250i)4-s + (−0.818 + 0.573i)5-s + 0.408·6-s + (−0.494 − 0.869i)7-s + (−0.250 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.541 − 0.454i)10-s + (−0.511 − 0.885i)11-s + (0.0747 + 0.278i)12-s + (−0.449 − 0.449i)13-s + (0.503 − 0.496i)14-s + (0.197 + 0.542i)15-s + (0.124 − 0.216i)16-s + (1.46 + 0.391i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.291091 - 0.443372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.291091 - 0.443372i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 3 | \( 1 + (-0.448 + 1.67i)T \) |
| 5 | \( 1 + (4.09 - 2.86i)T \) |
| 7 | \( 1 + (3.46 + 6.08i)T \) |
good | 11 | \( 1 + (5.62 + 9.74i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (5.84 + 5.84i)T + 169iT^{2} \) |
| 17 | \( 1 + (-24.8 - 6.65i)T + (250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (27.5 + 15.8i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (39.8 - 10.6i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 8.29iT - 841T^{2} \) |
| 31 | \( 1 + (-6.00 - 10.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-6.32 - 23.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 42.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (37.4 + 37.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-9.60 - 35.8i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (2.96 - 11.0i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-52.5 + 30.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-18.8 + 32.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (78.8 + 21.1i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 1.19T + 5.04e3T^{2} \) |
| 73 | \( 1 + (18.5 - 69.0i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (71.0 + 41.0i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-103. - 103. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (12.6 + 7.30i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-24.6 + 24.6i)T - 9.40e3iT^{2} \) |
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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
−12.05615840174873935913419068058, −10.83376860419125288049694057617, −9.954129571240342776146489596477, −8.284119182368596876370589266542, −7.78396363817158108461564054233, −6.79619304000068458988590453506, −5.82632830367674243460302303593, −4.12204864405124740807761048578, −3.03868798186374971567182046948, −0.26357765092275244725524742312,
2.27697072354204286611592098626, 3.75803844861075953954089829823, 4.73241606801513804695102726015, 5.90198840647969335455222493521, 7.71237436878018127318452154684, 8.649627251355863558242500758653, 9.706320686211004810200035971325, 10.34824408194547853996553278202, 11.81221578546421606129995908806, 12.24411568478353845687558511062