| L(s) = 1 | + (−45.2 − 78.3i)2-s + (1.03e3 + 597. i)3-s + (−4.09e3 + 7.09e3i)4-s + (5.10e3 − 2.94e3i)5-s − 1.08e5i·6-s + (−3.82e5 + 7.29e5i)7-s + 7.41e5·8-s + (−1.67e6 − 2.90e6i)9-s + (−4.62e5 − 2.66e5i)10-s + (1.25e7 − 2.17e7i)11-s + (−8.47e6 + 4.89e6i)12-s + 2.83e6i·13-s + (7.44e7 − 2.98e6i)14-s + 7.04e6·15-s + (−3.35e7 − 5.81e7i)16-s + (−5.15e8 − 2.97e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.473 + 0.273i)3-s + (−0.249 + 0.433i)4-s + (0.0653 − 0.0377i)5-s − 0.386i·6-s + (−0.464 + 0.885i)7-s + 0.353·8-s + (−0.350 − 0.607i)9-s + (−0.0462 − 0.0266i)10-s + (0.645 − 1.11i)11-s + (−0.236 + 0.136i)12-s + 0.0452i·13-s + (0.706 − 0.0283i)14-s + 0.0412·15-s + (−0.125 − 0.216i)16-s + (−1.25 − 0.724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.0418424 - 0.500440i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0418424 - 0.500440i\) |
| \(L(8)\) |
|
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 | \( 1 + (3.82e5 - 7.29e5i)T \) |
| good | 3 | \( 1 + (-1.03e3 - 597. i)T + (2.39e6 + 4.14e6i)T^{2} \) |
| 5 | \( 1 + (-5.10e3 + 2.94e3i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 11 | \( 1 + (-1.25e7 + 2.17e7i)T + (-1.89e14 - 3.28e14i)T^{2} \) |
| 13 | \( 1 - 2.83e6iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (5.15e8 + 2.97e8i)T + (8.41e16 + 1.45e17i)T^{2} \) |
| 19 | \( 1 + (9.77e8 - 5.64e8i)T + (3.99e17 - 6.91e17i)T^{2} \) |
| 23 | \( 1 + (2.11e9 + 3.66e9i)T + (-5.79e18 + 1.00e19i)T^{2} \) |
| 29 | \( 1 + 6.86e9T + 2.97e20T^{2} \) |
| 31 | \( 1 + (1.47e10 + 8.53e9i)T + (3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 + (-1.60e10 - 2.78e10i)T + (-4.50e21 + 7.80e21i)T^{2} \) |
| 41 | \( 1 + 2.21e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 2.95e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + (-1.50e11 + 8.70e10i)T + (1.28e23 - 2.22e23i)T^{2} \) |
| 53 | \( 1 + (2.74e11 - 4.75e11i)T + (-6.89e23 - 1.19e24i)T^{2} \) |
| 59 | \( 1 + (3.41e12 + 1.96e12i)T + (3.09e24 + 5.36e24i)T^{2} \) |
| 61 | \( 1 + (1.68e12 - 9.72e11i)T + (4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (1.21e12 - 2.09e12i)T + (-1.83e25 - 3.18e25i)T^{2} \) |
| 71 | \( 1 - 1.77e13T + 8.27e25T^{2} \) |
| 73 | \( 1 + (-6.99e12 - 4.03e12i)T + (6.10e25 + 1.05e26i)T^{2} \) |
| 79 | \( 1 + (-5.14e12 - 8.91e12i)T + (-1.84e26 + 3.19e26i)T^{2} \) |
| 83 | \( 1 - 4.46e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 + (3.18e13 - 1.83e13i)T + (9.78e26 - 1.69e27i)T^{2} \) |
| 97 | \( 1 + 1.12e14iT - 6.52e27T^{2} \) |
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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
−15.46560628820401182682724775750, −14.02671746519113290575363892462, −12.46024144235761685712081523971, −11.15992055736967736602425884221, −9.347424732238065791353942954789, −8.605941121178294774016851621917, −6.17605363514877408012447489113, −3.78688060313260603067512132723, −2.36513306573165323237428106939, −0.19022937715260855038714366863,
1.92325815960845896350303976343, 4.28617882164336182541084275136, 6.50031396182578914303022109627, 7.75516597597127337426773369511, 9.284026978458038247473710815525, 10.77466296114657655659638244574, 12.97541060055346692705455824141, 14.09799210766311983491794403996, 15.38794981916878903429979905897, 16.88557201418545533266641341409
