| L(s) = 1 | + 1.41·7-s − 6.37i·11-s − 3.54·13-s + 3.92·17-s + 1.27·19-s + 6.28i·23-s − 9.00·29-s − 3.92i·31-s + 2.51·37-s − 5.27i·41-s − 1.55i·43-s − 9.73i·47-s − 5·49-s − 5.55i·53-s + 0.313i·59-s + ⋯ |
| L(s) = 1 | + 0.534·7-s − 1.92i·11-s − 0.982·13-s + 0.952·17-s + 0.292·19-s + 1.31i·23-s − 1.67·29-s − 0.705i·31-s + 0.413·37-s − 0.823i·41-s − 0.237i·43-s − 1.42i·47-s − 0.714·49-s − 0.763i·53-s + 0.0408i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012474996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012474996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 6.37iT - 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 - 3.92T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 6.28iT - 23T^{2} \) |
| 29 | \( 1 + 9.00T + 29T^{2} \) |
| 31 | \( 1 + 3.92iT - 31T^{2} \) |
| 37 | \( 1 - 2.51T + 37T^{2} \) |
| 41 | \( 1 + 5.27iT - 41T^{2} \) |
| 43 | \( 1 + 1.55iT - 43T^{2} \) |
| 47 | \( 1 + 9.73iT - 47T^{2} \) |
| 53 | \( 1 + 5.55iT - 53T^{2} \) |
| 59 | \( 1 - 0.313iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 7.00iT - 67T^{2} \) |
| 71 | \( 1 + 0.990T + 71T^{2} \) |
| 73 | \( 1 - 12.0iT - 73T^{2} \) |
| 79 | \( 1 + 8.18iT - 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 + 0.386iT - 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−7.59976917008668856728024355176, −7.18646486890477842516667880505, −5.96801134965998878044767882472, −5.59612893625230404154037352485, −5.00186604125565790859909658345, −3.79060508815061752600663673080, −3.38231145964530390355014428722, −2.35974601817374221999171781179, −1.33046794068446672422995208282, −0.24169496041735158196782081601,
1.34410274441665682760457962129, 2.13679070038538337547645172183, 2.93911781290695069822126790080, 4.05820395799073134890072573119, 4.78590309444782465126389518645, 5.12262719877715450656029326381, 6.17593399073173064947368204412, 6.94168031148142188020376805656, 7.67889241067010748494950964028, 7.83948804100899595177777869835
