L(s) = 1 | + (−22.6 − 39.1i)2-s + (−745. − 430. i)3-s + (−1.02e3 + 1.77e3i)4-s + (−7.53e3 + 4.34e3i)5-s + 3.89e4i·6-s + (−9.37e4 + 7.10e4i)7-s + 9.26e4·8-s + (1.05e5 + 1.82e5i)9-s + (3.40e5 + 1.96e5i)10-s + (2.57e5 − 4.46e5i)11-s + (1.52e6 − 8.82e5i)12-s − 2.74e6i·13-s + (4.90e6 + 2.06e6i)14-s + 7.49e6·15-s + (−2.09e6 − 3.63e6i)16-s + (5.50e6 + 3.17e6i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−1.02 − 0.590i)3-s + (−0.249 + 0.433i)4-s + (−0.482 + 0.278i)5-s + 0.835i·6-s + (−0.797 + 0.603i)7-s + 0.353·8-s + (0.198 + 0.343i)9-s + (0.340 + 0.196i)10-s + (0.145 − 0.252i)11-s + (0.511 − 0.295i)12-s − 0.569i·13-s + (0.651 + 0.274i)14-s + 0.657·15-s + (−0.125 − 0.216i)16-s + (0.228 + 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(0.628997 - 0.0860986i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.628997 - 0.0860986i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (22.6 + 39.1i)T \) |
| 7 | \( 1 + (9.37e4 - 7.10e4i)T \) |
good | 3 | \( 1 + (745. + 430. i)T + (2.65e5 + 4.60e5i)T^{2} \) |
| 5 | \( 1 + (7.53e3 - 4.34e3i)T + (1.22e8 - 2.11e8i)T^{2} \) |
| 11 | \( 1 + (-2.57e5 + 4.46e5i)T + (-1.56e12 - 2.71e12i)T^{2} \) |
| 13 | \( 1 + 2.74e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (-5.50e6 - 3.17e6i)T + (2.91e14 + 5.04e14i)T^{2} \) |
| 19 | \( 1 + (-6.71e7 + 3.87e7i)T + (1.10e15 - 1.91e15i)T^{2} \) |
| 23 | \( 1 + (-5.20e7 - 9.01e7i)T + (-1.09e16 + 1.89e16i)T^{2} \) |
| 29 | \( 1 + 1.41e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (6.99e8 + 4.03e8i)T + (3.93e17 + 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-1.86e9 - 3.23e9i)T + (-3.29e18 + 5.70e18i)T^{2} \) |
| 41 | \( 1 - 4.75e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 8.94e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (-1.79e10 + 1.03e10i)T + (5.80e19 - 1.00e20i)T^{2} \) |
| 53 | \( 1 + (1.17e10 - 2.03e10i)T + (-2.45e20 - 4.25e20i)T^{2} \) |
| 59 | \( 1 + (1.83e9 + 1.05e9i)T + (8.89e20 + 1.54e21i)T^{2} \) |
| 61 | \( 1 + (7.28e9 - 4.20e9i)T + (1.32e21 - 2.29e21i)T^{2} \) |
| 67 | \( 1 + (-3.27e10 + 5.66e10i)T + (-4.09e21 - 7.08e21i)T^{2} \) |
| 71 | \( 1 + 2.46e11T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-8.87e10 - 5.12e10i)T + (1.14e22 + 1.98e22i)T^{2} \) |
| 79 | \( 1 + (-1.38e11 - 2.39e11i)T + (-2.95e22 + 5.11e22i)T^{2} \) |
| 83 | \( 1 + 8.18e10iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (4.69e10 - 2.71e10i)T + (1.23e23 - 2.13e23i)T^{2} \) |
| 97 | \( 1 + 3.42e11iT - 6.93e23T^{2} \) |
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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
−16.87133172308591403608187757674, −15.45759648096244130313796179827, −13.21993069500387021305651623578, −12.03681526296656914295017155693, −11.15428302363438231645544365097, −9.387212589470366182809730114405, −7.35708089917689496877017078052, −5.69831389114444441044651627134, −3.15543116150618638661228648411, −0.841148416294821185080732694701,
0.53577481698535349819061580682, 4.17721926014792828702411686883, 5.77980199090737286715925252292, 7.39235071609796546744447494082, 9.446504018525915515807143404194, 10.73257037130932244240006721365, 12.25352147874314529891973082888, 14.14903509009840311326323419605, 16.05369174522038814838858168174, 16.35556560076624160409421765623