| L(s) = 1 | + (19.5 − 11.3i)2-s + (255. − 443. i)4-s + (1.21e3 + 698. i)5-s + (7.82e3 + 1.35e4i)7-s − 1.15e4i·8-s + 3.16e4·10-s + (−1.94e5 + 1.12e5i)11-s + (−8.64e4 + 1.49e5i)13-s + (3.06e5 + 1.77e5i)14-s + (−1.31e5 − 2.27e5i)16-s + 6.84e5i·17-s + 1.00e6·19-s + (6.19e5 − 3.57e5i)20-s + (−2.53e6 + 4.39e6i)22-s + (−1.70e6 − 9.87e5i)23-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (0.465 + 0.806i)7-s − 0.353i·8-s + 0.316·10-s + (−1.20 + 0.696i)11-s + (−0.232 + 0.403i)13-s + (0.570 + 0.329i)14-s + (−0.125 − 0.216i)16-s + 0.482i·17-s + 0.404·19-s + (0.193 − 0.111i)20-s + (−0.492 + 0.853i)22-s + (−0.265 − 0.153i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.2353119817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2353119817\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-19.5 + 11.3i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.21e3 - 698. i)T \) |
| good | 7 | \( 1 + (-7.82e3 - 1.35e4i)T + (-1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 + (1.94e5 - 1.12e5i)T + (1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 + (8.64e4 - 1.49e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 - 6.84e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 1.00e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (1.70e6 + 9.87e5i)T + (2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (1.06e7 - 6.12e6i)T + (2.10e14 - 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-8.29e6 + 1.43e7i)T + (-4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 + 7.30e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (1.57e8 + 9.07e7i)T + (6.71e15 + 1.16e16i)T^{2} \) |
| 43 | \( 1 + (7.19e7 + 1.24e8i)T + (-1.08e16 + 1.87e16i)T^{2} \) |
| 47 | \( 1 + (6.39e7 - 3.69e7i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 + 1.54e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (3.56e8 + 2.06e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-5.82e8 - 1.00e9i)T + (-3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-9.22e8 + 1.59e9i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 + 3.30e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 6.65e8T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-2.00e9 - 3.47e9i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-4.94e9 + 2.85e9i)T + (7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 + 9.92e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (3.64e9 + 6.31e9i)T + (-3.68e19 + 6.38e19i)T^{2} \) |
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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
−9.971593823419459515396319194718, −8.860374637864506126507209050310, −7.73305713624425952686241138549, −6.63910447145840089339982091908, −5.45548208492852912977405635246, −4.92094102670814773863849307098, −3.52886676131880305381077344394, −2.30886271632313435918387573303, −1.78883605734254776996431932321, −0.03164599669777026033657151269,
1.20475716959833430721946610455, 2.57093470960159414697876498838, 3.59179170554665490957275437227, 4.90336481151108579589699196822, 5.44250361766568605178862140989, 6.68441532337564347309833918546, 7.71442127035821927674129938488, 8.374984506227237943427957118038, 9.787155931141096923784314486351, 10.66050793444652878593196291971
