| L(s) = 1 | + (4.89 − 2.82i)2-s + (−14.9 + 22.4i)3-s + (15.9 − 27.7i)4-s + (202. + 116. i)5-s + (−9.58 + 152. i)6-s + (95.5 + 165. i)7-s − 181. i·8-s + (−282. − 672. i)9-s + 1.32e3·10-s + (−673. + 388. i)11-s + (384. + 773. i)12-s + (45.5 − 78.9i)13-s + (936. + 540. i)14-s + (−5.64e3 + 2.80e3i)15-s + (−512. − 886. i)16-s − 7.04e3i·17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.553 + 0.832i)3-s + (0.249 − 0.433i)4-s + (1.61 + 0.934i)5-s + (−0.0443 + 0.705i)6-s + (0.278 + 0.482i)7-s − 0.353i·8-s + (−0.387 − 0.921i)9-s + 1.32·10-s + (−0.505 + 0.291i)11-s + (0.222 + 0.447i)12-s + (0.0207 − 0.0359i)13-s + (0.341 + 0.197i)14-s + (−1.67 + 0.830i)15-s + (−0.125 − 0.216i)16-s − 1.43i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.98741 + 0.584998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98741 + 0.584998i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 + 2.82i)T \) |
| 3 | \( 1 + (14.9 - 22.4i)T \) |
| good | 5 | \( 1 + (-202. - 116. i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-95.5 - 165. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (673. - 388. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-45.5 + 78.9i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 7.04e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 2.73e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.72e4 + 9.94e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.71e4 + 1.56e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-6.17e3 + 1.06e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 2.79e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (3.74e4 + 2.16e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-1.92e4 - 3.33e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.43e5 - 8.30e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 5.47e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (1.41e4 + 8.14e3i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.94e4 - 5.09e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.47e5 - 2.56e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.57e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 8.02e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-1.88e5 - 3.26e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-7.33e5 + 4.23e5i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + 1.12e3iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-6.75e5 - 1.16e6i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−17.69116123045969451275203970486, −15.99699330801760358346650753135, −14.66190269839696483640443271229, −13.70063761268388598062152130672, −11.87379381084840535983161764829, −10.47026477636543706282173826006, −9.623049250838380313456543442972, −6.34625458706468386567629492009, −5.12952550875453119040663468064, −2.60353758399536868074866720700,
1.64997347428988153635377497744, 5.19218039057125298158701569428, 6.29300855829618507063050984240, 8.252838173412415976424399969281, 10.37389348947025042140834347808, 12.28653500730699933857477124867, 13.34860981476329313118283518343, 14.03467266089762857567130013832, 16.27074516525627931739407220420, 17.29139922706735846707004970736
