| L(s) = 1 | + (22.6 + 39.1i)2-s + (−1.16e3 − 670. i)3-s + (−1.02e3 + 1.77e3i)4-s + (1.42e4 − 8.21e3i)5-s − 6.06e4i·6-s + (−9.60e4 − 6.80e4i)7-s − 9.26e4·8-s + (6.33e5 + 1.09e6i)9-s + (6.43e5 + 3.71e5i)10-s + (−4.64e5 + 8.03e5i)11-s + (2.37e6 − 1.37e6i)12-s + 8.24e6i·13-s + (4.92e5 − 5.30e6i)14-s − 2.20e7·15-s + (−2.09e6 − 3.63e6i)16-s + (1.17e7 + 6.77e6i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−1.59 − 0.919i)3-s + (−0.249 + 0.433i)4-s + (0.910 − 0.525i)5-s − 1.30i·6-s + (−0.816 − 0.578i)7-s − 0.353·8-s + (1.19 + 2.06i)9-s + (0.643 + 0.371i)10-s + (−0.261 + 0.453i)11-s + (0.796 − 0.459i)12-s + 1.70i·13-s + (0.0654 − 0.704i)14-s − 1.93·15-s + (−0.125 − 0.216i)16-s + (0.485 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0340 - 0.999i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.613470 + 0.634705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.613470 + 0.634705i\) |
| \(L(7)\) |
|
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 | \( 1 + (9.60e4 + 6.80e4i)T \) |
| good | 3 | \( 1 + (1.16e3 + 670. i)T + (2.65e5 + 4.60e5i)T^{2} \) |
| 5 | \( 1 + (-1.42e4 + 8.21e3i)T + (1.22e8 - 2.11e8i)T^{2} \) |
| 11 | \( 1 + (4.64e5 - 8.03e5i)T + (-1.56e12 - 2.71e12i)T^{2} \) |
| 13 | \( 1 - 8.24e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (-1.17e7 - 6.77e6i)T + (2.91e14 + 5.04e14i)T^{2} \) |
| 19 | \( 1 + (-3.88e7 + 2.24e7i)T + (1.10e15 - 1.91e15i)T^{2} \) |
| 23 | \( 1 + (-3.53e7 - 6.12e7i)T + (-1.09e16 + 1.89e16i)T^{2} \) |
| 29 | \( 1 + 5.34e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (-1.14e9 - 6.62e8i)T + (3.93e17 + 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-5.25e8 - 9.10e8i)T + (-3.29e18 + 5.70e18i)T^{2} \) |
| 41 | \( 1 - 1.78e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 4.20e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (1.50e10 - 8.68e9i)T + (5.80e19 - 1.00e20i)T^{2} \) |
| 53 | \( 1 + (-2.46e9 + 4.27e9i)T + (-2.45e20 - 4.25e20i)T^{2} \) |
| 59 | \( 1 + (-7.90e9 - 4.56e9i)T + (8.89e20 + 1.54e21i)T^{2} \) |
| 61 | \( 1 + (-1.85e10 + 1.07e10i)T + (1.32e21 - 2.29e21i)T^{2} \) |
| 67 | \( 1 + (3.21e10 - 5.56e10i)T + (-4.09e21 - 7.08e21i)T^{2} \) |
| 71 | \( 1 + 9.11e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (1.80e11 + 1.04e11i)T + (1.14e22 + 1.98e22i)T^{2} \) |
| 79 | \( 1 + (-1.52e11 - 2.64e11i)T + (-2.95e22 + 5.11e22i)T^{2} \) |
| 83 | \( 1 - 2.37e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (-9.71e10 + 5.60e10i)T + (1.23e23 - 2.13e23i)T^{2} \) |
| 97 | \( 1 + 9.61e11iT - 6.93e23T^{2} \) |
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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
−16.91412099620368509912832158449, −16.22852933033215587465430526062, −13.71120312737910349071709300264, −12.92659707271044182088409048391, −11.65799259772940031357856079446, −9.755523258474673604526027144322, −7.16459412858708722094289151799, −6.20152697540095666767364848598, −4.88462357191465809689086627040, −1.36972923895735790277844385159,
0.44603283081291402619789715413, 3.17599047292729537326060328830, 5.37849871479113949588383910412, 6.08013698831233191998622225744, 9.805751221504109357698813982880, 10.41348253075331098428989149803, 11.81331290313244282788581179122, 13.10249192455039378159913782947, 15.09639030674418138083829652977, 16.28563518493374628396342774652
