| L(s) = 1 | + 2.75i·3-s + (2.54 + 4.30i)5-s + 3.84·7-s + 1.41·9-s + 6.19·11-s − 16.1·13-s + (−11.8 + 7.01i)15-s − 5.20i·17-s − 36.2·19-s + 10.6i·21-s + 22.0·23-s + (−12.0 + 21.9i)25-s + 28.6i·27-s + 20.0i·29-s + 26.4i·31-s + ⋯ |
| L(s) = 1 | + 0.918i·3-s + (0.509 + 0.860i)5-s + 0.549·7-s + 0.157·9-s + 0.562·11-s − 1.23·13-s + (−0.789 + 0.467i)15-s − 0.306i·17-s − 1.90·19-s + 0.504i·21-s + 0.958·23-s + (−0.480 + 0.876i)25-s + 1.06i·27-s + 0.690i·29-s + 0.852i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.651302847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.651302847\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.54 - 4.30i)T \) |
| good | 3 | \( 1 - 2.75iT - 9T^{2} \) |
| 7 | \( 1 - 3.84T + 49T^{2} \) |
| 11 | \( 1 - 6.19T + 121T^{2} \) |
| 13 | \( 1 + 16.1T + 169T^{2} \) |
| 17 | \( 1 + 5.20iT - 289T^{2} \) |
| 19 | \( 1 + 36.2T + 361T^{2} \) |
| 23 | \( 1 - 22.0T + 529T^{2} \) |
| 29 | \( 1 - 20.0iT - 841T^{2} \) |
| 31 | \( 1 - 26.4iT - 961T^{2} \) |
| 37 | \( 1 + 69.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 25.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 39.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 27.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 54.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 107. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 37.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 97.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 6.40iT - 9.40e3T^{2} \) |
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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
−9.920988823709390074305702437066, −9.200216698074638877584134886003, −8.439276053856577167094970104714, −7.05785603478001421877190011382, −6.83492547055476113680182264411, −5.45724644532830480245777583852, −4.75584868265430379396577372697, −3.86621429970129235614906653088, −2.77434254112836508878712207093, −1.69218950264442558229230477410,
0.44837598788045507279297925579, 1.70407882511247889519426705425, 2.28898403926434295542501716099, 4.11090720216897815530790650083, 4.82155423076572412103718086350, 5.82712082599960228712791687785, 6.71750744561033076751641363314, 7.40908960340013629662364562655, 8.368975420399546048469231193841, 8.902637478856443757098270099145
