| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 2i·5-s + 2.82i·8-s − 2.82·10-s + 2.82i·11-s + 12.7·13-s + 4.00·16-s + 22i·17-s − 5.65·19-s + 4.00i·20-s + 4.00·22-s − 2.82i·23-s + 21·25-s − 18i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.400i·5-s + 0.353i·8-s − 0.282·10-s + 0.257i·11-s + 0.979·13-s + 0.250·16-s + 1.29i·17-s − 0.297·19-s + 0.200i·20-s + 0.181·22-s − 0.122i·23-s + 0.839·25-s − 0.692i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.785453127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.785453127\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 25T^{2} \) |
| 11 | \( 1 - 2.82iT - 121T^{2} \) |
| 13 | \( 1 - 12.7T + 169T^{2} \) |
| 17 | \( 1 - 22iT - 289T^{2} \) |
| 19 | \( 1 + 5.65T + 361T^{2} \) |
| 23 | \( 1 + 2.82iT - 529T^{2} \) |
| 29 | \( 1 - 35.3iT - 841T^{2} \) |
| 31 | \( 1 + 33.9T + 961T^{2} \) |
| 37 | \( 1 - 64T + 1.36e3T^{2} \) |
| 41 | \( 1 + 20iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44T + 1.84e3T^{2} \) |
| 47 | \( 1 + 68iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 100iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 52.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 120T + 4.48e3T^{2} \) |
| 71 | \( 1 + 8.48iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 92T + 6.24e3T^{2} \) |
| 83 | \( 1 + 112iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 20iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 26.8T + 9.40e3T^{2} \) |
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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
−10.00547031378230015936580832002, −8.874836262127587378111643916346, −8.573437046482523142514531420503, −7.41036329961863486728704744333, −6.26656363743895845733921610655, −5.36074127679982741914115330568, −4.26405032177997103244180749411, −3.48434341800496029820808240195, −2.10069955843459331158687924849, −0.983934921236531911442892866607,
0.792999614294037286584586582458, 2.60140568911097692391326393597, 3.75923726436234364632360657169, 4.79013568905701027944009123222, 5.85892779683274785563746997203, 6.53270812195965430283299446912, 7.46638228430010642778196845327, 8.197505043876838885171346721139, 9.170657549307608869558746810075, 9.773172567125420193964807802980
