| L(s) = 1 | + (6.65 − 9.15i)2-s + (−43.1 + 132. i)3-s + (−39.5 − 121. i)4-s + (371. − 269. i)5-s + (929. + 1.27e3i)6-s + (−2.98e3 + 968. i)7-s + (−1.37e3 − 447. i)8-s + (−1.04e4 − 7.62e3i)9-s − 5.19e3i·10-s + (−1.43e4 − 2.71e3i)11-s + 1.78e4·12-s + (−1.54e4 + 2.12e4i)13-s + (−1.09e4 + 3.37e4i)14-s + (1.98e4 + 6.10e4i)15-s + (−1.32e4 + 9.63e3i)16-s + (−3.79e4 − 5.21e4i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (−0.533 + 1.64i)3-s + (−0.154 − 0.475i)4-s + (0.594 − 0.431i)5-s + (0.717 + 0.987i)6-s + (−1.24 + 0.403i)7-s + (−0.336 − 0.109i)8-s + (−1.59 − 1.16i)9-s − 0.519i·10-s + (−0.982 − 0.185i)11-s + 0.862·12-s + (−0.540 + 0.743i)13-s + (−0.285 + 0.877i)14-s + (0.391 + 1.20i)15-s + (−0.202 + 0.146i)16-s + (−0.453 − 0.624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0621038 + 0.482477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0621038 + 0.482477i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-6.65 + 9.15i)T \) |
| 11 | \( 1 + (1.43e4 + 2.71e3i)T \) |
| good | 3 | \( 1 + (43.1 - 132. i)T + (-5.30e3 - 3.85e3i)T^{2} \) |
| 5 | \( 1 + (-371. + 269. i)T + (1.20e5 - 3.71e5i)T^{2} \) |
| 7 | \( 1 + (2.98e3 - 968. i)T + (4.66e6 - 3.38e6i)T^{2} \) |
| 13 | \( 1 + (1.54e4 - 2.12e4i)T + (-2.52e8 - 7.75e8i)T^{2} \) |
| 17 | \( 1 + (3.79e4 + 5.21e4i)T + (-2.15e9 + 6.63e9i)T^{2} \) |
| 19 | \( 1 + (-1.41e5 - 4.58e4i)T + (1.37e10 + 9.98e9i)T^{2} \) |
| 23 | \( 1 + 1.81e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (2.48e5 - 8.07e4i)T + (4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-1.18e6 - 8.59e5i)T + (2.63e11 + 8.11e11i)T^{2} \) |
| 37 | \( 1 + (8.01e5 + 2.46e6i)T + (-2.84e12 + 2.06e12i)T^{2} \) |
| 41 | \( 1 + (-3.13e6 - 1.01e6i)T + (6.45e12 + 4.69e12i)T^{2} \) |
| 43 | \( 1 - 6.20e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (1.90e6 - 5.86e6i)T + (-1.92e13 - 1.39e13i)T^{2} \) |
| 53 | \( 1 + (8.14e6 + 5.91e6i)T + (1.92e13 + 5.92e13i)T^{2} \) |
| 59 | \( 1 + (5.33e6 + 1.64e7i)T + (-1.18e14 + 8.63e13i)T^{2} \) |
| 61 | \( 1 + (-5.56e6 - 7.65e6i)T + (-5.92e13 + 1.82e14i)T^{2} \) |
| 67 | \( 1 + 9.29e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + (-1.53e7 + 1.11e7i)T + (1.99e14 - 6.14e14i)T^{2} \) |
| 73 | \( 1 + (2.04e7 - 6.65e6i)T + (6.52e14 - 4.74e14i)T^{2} \) |
| 79 | \( 1 + (1.55e7 - 2.14e7i)T + (-4.68e14 - 1.44e15i)T^{2} \) |
| 83 | \( 1 + (1.74e7 + 2.40e7i)T + (-6.95e14 + 2.14e15i)T^{2} \) |
| 89 | \( 1 + 1.07e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (7.62e7 + 5.54e7i)T + (2.42e15 + 7.45e15i)T^{2} \) |
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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
−16.20720708318129481268965615536, −15.84708787991768462184020243483, −14.16307278031416593019751034347, −12.68735885360955098450320055661, −11.28307110903885793037900640242, −9.844193370925214471871608841269, −9.401580415085600547072271409299, −5.90122495886931752821524610539, −4.76759251615457594353730763162, −3.04449869600538338137551478034,
0.20678711744201538062752455614, 2.61635746540510792563566555913, 5.71929993634626273502505066856, 6.72312931976512971553693519169, 7.80390794041197507623216275060, 10.19097278474166220762748872802, 12.12502699715677034291553895327, 13.18790363337116023497262636296, 13.72189715863676204878550862096, 15.59858695744385457073649063644
