| L(s) = 1 | + (45.2 − 78.3i)2-s + (1.29e3 − 749. i)3-s + (−4.09e3 − 7.09e3i)4-s + (−6.75e4 − 3.89e4i)5-s − 1.35e5i·6-s + (−3.46e5 + 7.47e5i)7-s − 7.41e5·8-s + (−1.26e6 + 2.19e6i)9-s + (−6.11e6 + 3.52e6i)10-s + (−4.94e6 − 8.57e6i)11-s + (−1.06e7 − 6.14e6i)12-s − 1.06e7i·13-s + (4.29e7 + 6.09e7i)14-s − 1.16e8·15-s + (−3.35e7 + 5.81e7i)16-s + (−2.56e8 + 1.47e8i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.593 − 0.342i)3-s + (−0.249 − 0.433i)4-s + (−0.864 − 0.498i)5-s − 0.484i·6-s + (−0.420 + 0.907i)7-s − 0.353·8-s + (−0.264 + 0.458i)9-s + (−0.611 + 0.352i)10-s + (−0.253 − 0.439i)11-s + (−0.296 − 0.171i)12-s − 0.170i·13-s + (0.407 + 0.578i)14-s − 0.684·15-s + (−0.125 + 0.216i)16-s + (−0.624 + 0.360i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.00261295 + 0.00433921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00261295 + 0.00433921i\) |
| \(L(8)\) |
|
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 | \( 1 + (3.46e5 - 7.47e5i)T \) |
| good | 3 | \( 1 + (-1.29e3 + 749. i)T + (2.39e6 - 4.14e6i)T^{2} \) |
| 5 | \( 1 + (6.75e4 + 3.89e4i)T + (3.05e9 + 5.28e9i)T^{2} \) |
| 11 | \( 1 + (4.94e6 + 8.57e6i)T + (-1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + 1.06e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (2.56e8 - 1.47e8i)T + (8.41e16 - 1.45e17i)T^{2} \) |
| 19 | \( 1 + (-3.85e8 - 2.22e8i)T + (3.99e17 + 6.91e17i)T^{2} \) |
| 23 | \( 1 + (1.70e9 - 2.95e9i)T + (-5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + 2.75e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + (5.44e9 - 3.14e9i)T + (3.78e20 - 6.55e20i)T^{2} \) |
| 37 | \( 1 + (-9.12e10 + 1.58e11i)T + (-4.50e21 - 7.80e21i)T^{2} \) |
| 41 | \( 1 + 1.84e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 1.16e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + (6.76e11 + 3.90e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 + (-6.00e11 - 1.03e12i)T + (-6.89e23 + 1.19e24i)T^{2} \) |
| 59 | \( 1 + (3.90e11 - 2.25e11i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (2.34e12 + 1.35e12i)T + (4.93e24 + 8.55e24i)T^{2} \) |
| 67 | \( 1 + (7.27e11 + 1.26e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 - 3.35e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + (-9.10e12 + 5.25e12i)T + (6.10e25 - 1.05e26i)T^{2} \) |
| 79 | \( 1 + (-1.16e13 + 2.01e13i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 - 5.19e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 + (2.25e13 + 1.29e13i)T + (9.78e26 + 1.69e27i)T^{2} \) |
| 97 | \( 1 - 5.20e13iT - 6.52e27T^{2} \) |
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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
−16.27235853426643643678861123782, −15.07073972466760169489503038025, −13.53372831640706068971923928471, −12.44350026932344441562530478710, −11.18281910765394421618558311731, −9.149721085623578302651491005219, −7.914838070212562362177109637032, −5.52353284916980907160400200958, −3.59827535725765613907698565371, −2.09800297584096273207199381292,
0.00150590488972926791058820847, 3.15661211061812712715001322753, 4.31350935406474326057709710405, 6.68234118078833129556279392047, 7.929085960611061644118219613864, 9.632242590277253268571964319307, 11.45438580522878333890570779695, 13.21099026431385294994713490467, 14.56250237539051471555407120176, 15.43619340251847013311195516850
