| L(s) = 1 | + (−2.44 + 1.41i)2-s + (6.85 − 5.82i)3-s + (3.99 − 6.92i)4-s + (37.0 + 21.3i)5-s + (−8.55 + 23.9i)6-s + (−19.8 − 34.3i)7-s + 22.6i·8-s + (13.0 − 79.9i)9-s − 120.·10-s + (−139. + 80.7i)11-s + (−12.9 − 70.8i)12-s + (−36.6 + 63.4i)13-s + (97.2 + 56.1i)14-s + (378. − 69.1i)15-s + (−32.0 − 55.4i)16-s + 65.0i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.762 − 0.647i)3-s + (0.249 − 0.433i)4-s + (1.48 + 0.854i)5-s + (−0.237 + 0.665i)6-s + (−0.404 − 0.701i)7-s + 0.353i·8-s + (0.161 − 0.986i)9-s − 1.20·10-s + (−1.15 + 0.667i)11-s + (−0.0898 − 0.491i)12-s + (−0.216 + 0.375i)13-s + (0.495 + 0.286i)14-s + (1.68 − 0.307i)15-s + (−0.125 − 0.216i)16-s + 0.224i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0124i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.26360 + 0.00789693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26360 + 0.00789693i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.44 - 1.41i)T \) |
| 3 | \( 1 + (-6.85 + 5.82i)T \) |
| good | 5 | \( 1 + (-37.0 - 21.3i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (19.8 + 34.3i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (139. - 80.7i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (36.6 - 63.4i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 65.0iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 163.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (185. + 107. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (563. - 325. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-633. + 1.09e3i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 278.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-48.2 - 27.8i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-932. - 1.61e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-1.10e3 + 636. i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 - 3.58e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-2.93e3 - 1.69e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (3.49e3 + 6.04e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (958. - 1.66e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 4.58e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.91e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (974. + 1.68e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-1.19e3 + 688. i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + 1.37e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-4.14e3 - 7.18e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−18.06997436573494588029145420382, −17.04512993358969649783780287610, −15.13384528124555762245357909419, −14.00572588854808056094326507741, −13.02427860822468016834152867658, −10.45023962820383349089138069180, −9.491684936447435553934235958703, −7.53858231447169443503710439509, −6.31380721354533049090847999039, −2.28589222508502348599037561037,
2.48999694692836780762156599905, 5.46950545142362500623511304573, 8.392897795354847585846403299203, 9.427511489645825418478135922192, 10.44296925859278060277533966493, 12.75350423828468765095059092499, 13.76462942885900577143963963955, 15.60927200947669009841076360068, 16.66220687114557178499390265274, 17.99056415480924181200363341213
