L(s) = 1 | + (−45.2 − 78.3i)2-s + (−3.57e3 − 2.06e3i)3-s + (−4.09e3 + 7.09e3i)4-s + (8.10e4 − 4.67e4i)5-s + 3.73e5i·6-s + (5.87e5 + 5.77e5i)7-s + 7.41e5·8-s + (6.11e6 + 1.05e7i)9-s + (−7.33e6 − 4.23e6i)10-s + (1.07e7 − 1.85e7i)11-s + (2.92e7 − 1.68e7i)12-s + 9.80e6i·13-s + (1.86e7 − 7.21e7i)14-s − 3.85e8·15-s + (−3.35e7 − 5.81e7i)16-s + (5.15e8 + 2.97e8i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−1.63 − 0.942i)3-s + (−0.249 + 0.433i)4-s + (1.03 − 0.599i)5-s + 1.33i·6-s + (0.712 + 0.701i)7-s + 0.353·8-s + (1.27 + 2.21i)9-s + (−0.733 − 0.423i)10-s + (0.550 − 0.952i)11-s + (0.816 − 0.471i)12-s + 0.156i·13-s + (0.177 − 0.684i)14-s − 2.25·15-s + (−0.125 − 0.216i)16-s + (1.25 + 0.725i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(0.791462 - 0.898781i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.791462 - 0.898781i\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (45.2 + 78.3i)T \) |
| 7 | \( 1 + (-5.87e5 - 5.77e5i)T \) |
good | 3 | \( 1 + (3.57e3 + 2.06e3i)T + (2.39e6 + 4.14e6i)T^{2} \) |
| 5 | \( 1 + (-8.10e4 + 4.67e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 11 | \( 1 + (-1.07e7 + 1.85e7i)T + (-1.89e14 - 3.28e14i)T^{2} \) |
| 13 | \( 1 - 9.80e6iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (-5.15e8 - 2.97e8i)T + (8.41e16 + 1.45e17i)T^{2} \) |
| 19 | \( 1 + (1.96e8 - 1.13e8i)T + (3.99e17 - 6.91e17i)T^{2} \) |
| 23 | \( 1 + (7.21e8 + 1.24e9i)T + (-5.79e18 + 1.00e19i)T^{2} \) |
| 29 | \( 1 - 1.54e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + (-2.91e10 - 1.68e10i)T + (3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 + (1.75e10 + 3.03e10i)T + (-4.50e21 + 7.80e21i)T^{2} \) |
| 41 | \( 1 - 9.80e10iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 3.91e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + (-6.95e11 + 4.01e11i)T + (1.28e23 - 2.22e23i)T^{2} \) |
| 53 | \( 1 + (-9.24e11 + 1.60e12i)T + (-6.89e23 - 1.19e24i)T^{2} \) |
| 59 | \( 1 + (9.16e11 + 5.29e11i)T + (3.09e24 + 5.36e24i)T^{2} \) |
| 61 | \( 1 + (5.05e11 - 2.91e11i)T + (4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (-5.19e12 + 8.99e12i)T + (-1.83e25 - 3.18e25i)T^{2} \) |
| 71 | \( 1 + 5.47e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + (-2.19e12 - 1.26e12i)T + (6.10e25 + 1.05e26i)T^{2} \) |
| 79 | \( 1 + (3.31e12 + 5.73e12i)T + (-1.84e26 + 3.19e26i)T^{2} \) |
| 83 | \( 1 + 1.05e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 + (2.07e13 - 1.19e13i)T + (9.78e26 - 1.69e27i)T^{2} \) |
| 97 | \( 1 - 1.03e14iT - 6.52e27T^{2} \) |
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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
−16.57625691775763727583684090543, −13.79626702280993023891255918811, −12.47216218911066502102321705154, −11.67161349452280729596333694880, −10.30753867745080989500312805769, −8.361092629351579040593172536351, −6.21952782274343982914919122660, −5.16655865906593087487633082006, −1.78966370199260471053543189647, −0.899263361815036445163496445790,
1.04165492530931547247547726871, 4.51189841116919996370008935947, 5.75639772158882780924640483180, 7.00557467586186044500717170325, 9.790069348465159731480898991873, 10.39314251533131651804017838619, 11.85819562260960303212050163730, 14.12014775464293727998000671508, 15.38554751931883828887043551150, 16.88148905181978476425529613027