| L(s) = 1 | + (−1.50 + 1.31i)2-s + (0.519 − 3.96i)4-s + (−3.47 − 6.02i)5-s + (2.29 − 3.97i)7-s + (4.45 + 6.64i)8-s + (13.1 + 4.46i)10-s + (−7.77 + 13.4i)11-s + (−19.3 + 11.1i)13-s + (1.79 + 9.00i)14-s + (−15.4 − 4.12i)16-s + 9.17i·17-s + 4.53i·19-s + (−25.6 + 10.6i)20-s + (−6.07 − 30.4i)22-s + (2.69 − 1.55i)23-s + ⋯ |
| L(s) = 1 | + (−0.751 + 0.659i)2-s + (0.129 − 0.991i)4-s + (−0.695 − 1.20i)5-s + (0.327 − 0.567i)7-s + (0.556 + 0.830i)8-s + (1.31 + 0.446i)10-s + (−0.706 + 1.22i)11-s + (−1.49 + 0.860i)13-s + (0.128 + 0.643i)14-s + (−0.966 − 0.257i)16-s + 0.539i·17-s + 0.238i·19-s + (−1.28 + 0.532i)20-s + (−0.276 − 1.38i)22-s + (0.117 − 0.0676i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0739i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00300613 + 0.0811766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00300613 + 0.0811766i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.50 - 1.31i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.47 + 6.02i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-2.29 + 3.97i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (7.77 - 13.4i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (19.3 - 11.1i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 9.17iT - 289T^{2} \) |
| 19 | \( 1 - 4.53iT - 361T^{2} \) |
| 23 | \( 1 + (-2.69 + 1.55i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-6.42 + 11.1i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (2.17 + 3.77i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 23.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (60.6 - 35.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (51.7 + 29.8i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (32.1 + 18.5i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 24.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (20.7 + 35.9i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-34.6 - 19.9i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-24.3 + 14.0i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 59.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + (0.557 - 0.965i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-39.8 + 69.0i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 10.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (72.6 - 125. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.42292110086977281086940837890, −11.64387150394242384829197157290, −10.25898296118762424770209493331, −9.592522378970894765196416817429, −8.407108991245920780344442322071, −7.69702270339289185270301197308, −6.82067577627752564158457516979, −5.01261984162951409057481682380, −4.53409069818284032165470086066, −1.77245403397385880801639960228,
0.05504025784280158865895910020, 2.59536772111803435236211839416, 3.31585214107209022120878476817, 5.17754453387076279746687114303, 6.89365766741819481132605112822, 7.75199948508189793371990995896, 8.564665828869008972691020241240, 9.898646607110278230559817017576, 10.71857726085660067498865178768, 11.43886654408467103263971766659
