| L(s) = 1 | + (−466. + 808. i)2-s + (−2.56e4 + 1.47e4i)3-s + (−3.04e5 − 5.28e5i)4-s + (1.26e6 + 7.29e5i)5-s − 2.76e7i·6-s + (3.62e7 + 1.77e7i)7-s + 3.24e8·8-s + (2.44e8 − 4.23e8i)9-s + (−1.18e9 + 6.81e8i)10-s + (−1.86e9 − 3.23e9i)11-s + (1.56e10 + 9.02e9i)12-s − 7.49e9i·13-s + (−3.12e10 + 2.10e10i)14-s − 4.31e10·15-s + (−7.17e10 + 1.24e11i)16-s + (4.27e10 − 2.46e10i)17-s + ⋯ |
| L(s) = 1 | + (−0.911 + 1.57i)2-s + (−1.30 + 0.751i)3-s + (−1.16 − 2.01i)4-s + (0.647 + 0.373i)5-s − 2.74i·6-s + (0.898 + 0.438i)7-s + 2.42·8-s + (0.630 − 1.09i)9-s + (−1.18 + 0.681i)10-s + (−0.792 − 1.37i)11-s + (3.03 + 1.74i)12-s − 0.707i·13-s + (−1.51 + 1.01i)14-s − 1.12·15-s + (−1.04 + 1.80i)16-s + (0.360 − 0.208i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0573 - 0.998i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.0573 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.487162 + 0.459967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487162 + 0.459967i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-3.62e7 - 1.77e7i)T \) |
| good | 2 | \( 1 + (466. - 808. i)T + (-1.31e5 - 2.27e5i)T^{2} \) |
| 3 | \( 1 + (2.56e4 - 1.47e4i)T + (1.93e8 - 3.35e8i)T^{2} \) |
| 5 | \( 1 + (-1.26e6 - 7.29e5i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 11 | \( 1 + (1.86e9 + 3.23e9i)T + (-2.77e18 + 4.81e18i)T^{2} \) |
| 13 | \( 1 + 7.49e9iT - 1.12e20T^{2} \) |
| 17 | \( 1 + (-4.27e10 + 2.46e10i)T + (7.03e21 - 1.21e22i)T^{2} \) |
| 19 | \( 1 + (-3.85e11 - 2.22e11i)T + (5.20e22 + 9.01e22i)T^{2} \) |
| 23 | \( 1 + (1.81e11 - 3.14e11i)T + (-1.62e24 - 2.80e24i)T^{2} \) |
| 29 | \( 1 + 1.15e13T + 2.10e26T^{2} \) |
| 31 | \( 1 + (-7.81e12 + 4.50e12i)T + (3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 + (-9.55e13 + 1.65e14i)T + (-8.44e27 - 1.46e28i)T^{2} \) |
| 41 | \( 1 - 1.01e14iT - 1.07e29T^{2} \) |
| 43 | \( 1 - 4.04e14T + 2.52e29T^{2} \) |
| 47 | \( 1 + (9.33e14 + 5.39e14i)T + (6.26e29 + 1.08e30i)T^{2} \) |
| 53 | \( 1 + (4.09e14 + 7.08e14i)T + (-5.44e30 + 9.42e30i)T^{2} \) |
| 59 | \( 1 + (-4.96e15 + 2.86e15i)T + (3.75e31 - 6.49e31i)T^{2} \) |
| 61 | \( 1 + (8.76e15 + 5.06e15i)T + (6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (-1.87e16 - 3.24e16i)T + (-3.70e32 + 6.41e32i)T^{2} \) |
| 71 | \( 1 - 4.25e16T + 2.10e33T^{2} \) |
| 73 | \( 1 + (-5.60e16 + 3.23e16i)T + (1.73e33 - 3.00e33i)T^{2} \) |
| 79 | \( 1 + (3.89e16 - 6.74e16i)T + (-7.18e33 - 1.24e34i)T^{2} \) |
| 83 | \( 1 - 1.31e17iT - 3.49e34T^{2} \) |
| 89 | \( 1 + (1.05e17 + 6.09e16i)T + (6.13e34 + 1.06e35i)T^{2} \) |
| 97 | \( 1 + 1.20e18iT - 5.77e35T^{2} \) |
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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
−17.80204190404516460509269317019, −16.57652758717008465816459491175, −15.66753146931953655748877934380, −14.17550086437819399871779344441, −11.06360181175916888991470562904, −9.830057592618625785199003625440, −7.975825916300218497681626581564, −5.83598930164509170521876994698, −5.39155815276248246027375159100, −0.59611179135781350701441263225,
1.00274081222844277889518209454, 1.93349913441184503759841450942, 4.88525412156540508252722244937, 7.53769468779295973423509156438, 9.702489851839038972398296712603, 11.07917523522622204469317822093, 12.11913941910602064654180008353, 13.29288873373283260302905160090, 16.99422975990923283313970075474, 17.72686855087598274673633139794
