| L(s) = 1 | − 1.62i·2-s + 1.36·4-s + 2.23i·5-s + 7.80·7-s − 8.70i·8-s + 3.62·10-s + 3.31i·11-s + 0.441·13-s − 12.6i·14-s − 8.66·16-s + 7.90i·17-s + 24.6·19-s + 3.05i·20-s + 5.38·22-s − 7.46i·23-s + ⋯ |
| L(s) = 1 | − 0.811i·2-s + 0.341·4-s + 0.447i·5-s + 1.11·7-s − 1.08i·8-s + 0.362·10-s + 0.301i·11-s + 0.0339·13-s − 0.904i·14-s − 0.541·16-s + 0.464i·17-s + 1.29·19-s + 0.152i·20-s + 0.244·22-s − 0.324i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.395328194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.395328194\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 11 | \( 1 - 3.31iT \) |
| good | 2 | \( 1 + 1.62iT - 4T^{2} \) |
| 7 | \( 1 - 7.80T + 49T^{2} \) |
| 13 | \( 1 - 0.441T + 169T^{2} \) |
| 17 | \( 1 - 7.90iT - 289T^{2} \) |
| 19 | \( 1 - 24.6T + 361T^{2} \) |
| 23 | \( 1 + 7.46iT - 529T^{2} \) |
| 29 | \( 1 - 22.2iT - 841T^{2} \) |
| 31 | \( 1 - 15.0T + 961T^{2} \) |
| 37 | \( 1 - 28.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 15.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 49.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 73.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 6.38iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 42.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 54.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2.26T + 4.48e3T^{2} \) |
| 71 | \( 1 + 62.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 32.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 107.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 33.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 9.85iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.5T + 9.40e3T^{2} \) |
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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
−10.68968411873450611333057315127, −10.06240329472867632986780224242, −8.970493150456620639316821407925, −7.77801940999577484614900300732, −7.08773969941112916659580871191, −5.91747024212329951978474360709, −4.67947729434122213728930986417, −3.49679808121969647703567617981, −2.34905919517187636289209031044, −1.23259988159205787834771104224,
1.30583596735573487656586156459, 2.76012394749710591610739626671, 4.48045104152428133396625096313, 5.36024375139976200264414822220, 6.17230166519206286258966200339, 7.47808003541101120426386491584, 7.88411097344665987324473889026, 8.837348780105886526132973557608, 9.855604008300834315683650919502, 11.25184223817963987019976345926
