| L(s) = 1 | + (−17.4 − 6.35i)2-s + (−127. − 107. i)4-s + (−280. + 1.58e3i)5-s + (610. − 512. i)7-s + (6.30e3 + 1.09e4i)8-s + (1.50e4 − 2.59e4i)10-s + (4.49e3 + 2.54e4i)11-s + (1.06e5 − 3.86e4i)13-s + (−1.39e4 + 5.06e3i)14-s + (−2.58e4 − 1.46e5i)16-s + (−6.42e4 + 1.11e5i)17-s + (−1.75e5 − 3.04e5i)19-s + (2.06e5 − 1.72e5i)20-s + (8.35e4 − 4.73e5i)22-s + (1.96e3 + 1.64e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.771 − 0.280i)2-s + (−0.249 − 0.209i)4-s + (−0.200 + 1.13i)5-s + (0.0961 − 0.0806i)7-s + (0.544 + 0.942i)8-s + (0.474 − 0.821i)10-s + (0.0925 + 0.524i)11-s + (1.03 − 0.375i)13-s + (−0.0968 + 0.0352i)14-s + (−0.0987 − 0.560i)16-s + (−0.186 + 0.323i)17-s + (−0.309 − 0.536i)19-s + (0.287 − 0.241i)20-s + (0.0760 − 0.431i)22-s + (0.00146 + 0.00122i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.345649 + 0.590237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.345649 + 0.590237i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (17.4 + 6.35i)T + (392. + 329. i)T^{2} \) |
| 5 | \( 1 + (280. - 1.58e3i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (-610. + 512. i)T + (7.00e6 - 3.97e7i)T^{2} \) |
| 11 | \( 1 + (-4.49e3 - 2.54e4i)T + (-2.21e9 + 8.06e8i)T^{2} \) |
| 13 | \( 1 + (-1.06e5 + 3.86e4i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (6.42e4 - 1.11e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.75e5 + 3.04e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.96e3 - 1.64e3i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (-3.90e6 - 1.42e6i)T + (1.11e13 + 9.32e12i)T^{2} \) |
| 31 | \( 1 + (4.10e6 + 3.44e6i)T + (4.59e12 + 2.60e13i)T^{2} \) |
| 37 | \( 1 + (-1.52e6 + 2.64e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-2.20e7 + 8.02e6i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-3.33e6 - 1.89e7i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (1.67e7 - 1.40e7i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + 8.11e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (3.01e7 - 1.70e8i)T + (-8.14e15 - 2.96e15i)T^{2} \) |
| 61 | \( 1 + (7.48e7 - 6.27e7i)T + (2.03e15 - 1.15e16i)T^{2} \) |
| 67 | \( 1 + (-2.17e8 + 7.92e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (1.41e8 - 2.45e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-2.29e8 - 3.97e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (5.29e8 + 1.92e8i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (6.29e7 + 2.28e7i)T + (1.43e17 + 1.20e17i)T^{2} \) |
| 89 | \( 1 + (2.71e8 + 4.69e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-1.23e7 - 7.03e7i)T + (-7.14e17 + 2.60e17i)T^{2} \) |
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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
−12.85043615253663272348784955861, −11.16307225872655975605520343099, −10.76375068513399559628657718401, −9.631953986895306300013693257324, −8.483466501788578128473723054893, −7.32502275294755885399620021760, −6.00321501227222310732550876340, −4.32934249624878045170180381564, −2.72505371874084125192854852653, −1.24240336678909573524192562746,
0.30138137668303891394576215019, 1.37787166097340610883951404169, 3.68268486884900204430973628745, 4.86075853321564060638842202652, 6.46126991106343218498337153625, 8.022185329507562147709477899184, 8.653228837553833862522526758559, 9.502451198933551253977537109570, 10.92503537345982541649652179200, 12.25159068775253815697603548098
