| L(s) = 1 | + (0.557 + 4.96i)5-s + 10.7i·7-s − 12.1i·11-s − 23.3i·13-s − 1.67·17-s + 0.234·19-s − 19.8·23-s + (−24.3 + 5.53i)25-s − 34.0i·29-s − 19.0·31-s + (−53.2 + 5.97i)35-s + 32.7i·37-s − 5.27i·41-s − 52.9i·43-s − 53.7·47-s + ⋯ |
| L(s) = 1 | + (0.111 + 0.993i)5-s + 1.53i·7-s − 1.10i·11-s − 1.79i·13-s − 0.0987·17-s + 0.0123·19-s − 0.863·23-s + (−0.975 + 0.221i)25-s − 1.17i·29-s − 0.615·31-s + (−1.52 + 0.170i)35-s + 0.884i·37-s − 0.128i·41-s − 1.23i·43-s − 1.14·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9434515740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9434515740\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.557 - 4.96i)T \) |
| good | 7 | \( 1 - 10.7iT - 49T^{2} \) |
| 11 | \( 1 + 12.1iT - 121T^{2} \) |
| 13 | \( 1 + 23.3iT - 169T^{2} \) |
| 17 | \( 1 + 1.67T + 289T^{2} \) |
| 19 | \( 1 - 0.234T + 361T^{2} \) |
| 23 | \( 1 + 19.8T + 529T^{2} \) |
| 29 | \( 1 + 34.0iT - 841T^{2} \) |
| 31 | \( 1 + 19.0T + 961T^{2} \) |
| 37 | \( 1 - 32.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 5.27iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 52.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 53.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 84.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 88.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 65.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 99.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 36.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 79.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 35.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 27.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 152. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 97.0iT - 9.40e3T^{2} \) |
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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
−8.908596004268588771875411552983, −8.239583455166677725980772538967, −7.58974730435798731554493165913, −6.30005336468035754147337082788, −5.87077548983621918345975320468, −5.20532980811727080121984666121, −3.59797184687239130877164401618, −2.92901376669619441174456170991, −2.10827777910494673171293386155, −0.25546939285736970359761774354,
1.22746020008195890808052207371, 2.01322974152932405593105180034, 3.82847132277545128717208598659, 4.34091519949284801026442804698, 5.01257970229750706273411110034, 6.30035673059794134252834432371, 7.11556975807362719133160061696, 7.64005601505891032503533112121, 8.721483779130827551838494073041, 9.435102770044926937170927675319
