| L(s) = 1 | + 9.23i·3-s − 11.1·5-s − 33.5i·7-s − 58.3·9-s − 103. i·15-s + 310.·21-s − 73.8i·23-s + 125.·25-s − 289. i·27-s − 306·29-s + 375. i·35-s − 460.·41-s − 563. i·43-s + 651.·45-s + 41.1i·47-s + ⋯ |
| L(s) = 1 | + 1.77i·3-s − 0.999·5-s − 1.81i·7-s − 2.15·9-s − 1.77i·15-s + 3.22·21-s − 0.669i·23-s + 1.00·25-s − 2.06i·27-s − 1.95·29-s + 1.81i·35-s − 1.75·41-s − 1.99i·43-s + 2.15·45-s + 0.127i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.257279 - 0.257279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.257279 - 0.257279i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 11.1T \) |
| good | 3 | \( 1 - 9.23iT - 27T^{2} \) |
| 7 | \( 1 + 33.5iT - 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 306T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 460.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 563. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 41.1iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 40.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.16iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 - 989. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−11.79421322137087531019754607750, −10.79596179798570009025476218345, −10.43086524135136490420810291785, −9.294884277794209984645886575814, −8.121616736501044980628185671036, −6.98578195471648706148523551539, −5.10593770879553400022617563211, −4.05964249130542937344421446193, −3.55265180026937262639531173105, −0.16759679735773697081094993538,
1.78298412618732499995550685063, 3.08780382429683217382433433786, 5.35982987533224370712558748595, 6.38507898824676997381134157602, 7.52858319916217016773528016669, 8.296471432672225401798581744118, 9.187789002479131000662147010356, 11.35534469946696054925723903055, 11.80030221658430394044630585935, 12.59591299524810132180713244891
