| L(s) = 1 | − 1.45·2-s + 5.64·3-s − 5.87·4-s − 5·5-s − 8.23·6-s − 17.6·7-s + 20.2·8-s + 4.87·9-s + 7.29·10-s − 11·11-s − 33.1·12-s − 0.138·13-s + 25.7·14-s − 28.2·15-s + 17.4·16-s + 28.7·17-s − 7.11·18-s − 19·19-s + 29.3·20-s − 99.5·21-s + 16.0·22-s − 148.·23-s + 114.·24-s + 25·25-s + 0.201·26-s − 124.·27-s + 103.·28-s + ⋯ |
| L(s) = 1 | − 0.515·2-s + 1.08·3-s − 0.733·4-s − 0.447·5-s − 0.560·6-s − 0.952·7-s + 0.894·8-s + 0.180·9-s + 0.230·10-s − 0.301·11-s − 0.797·12-s − 0.00294·13-s + 0.491·14-s − 0.485·15-s + 0.272·16-s + 0.410·17-s − 0.0931·18-s − 0.229·19-s + 0.328·20-s − 1.03·21-s + 0.155·22-s − 1.34·23-s + 0.972·24-s + 0.200·25-s + 0.00152·26-s − 0.890·27-s + 0.698·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.006614136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006614136\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 1.45T + 8T^{2} \) |
| 3 | \( 1 - 5.64T + 27T^{2} \) |
| 7 | \( 1 + 17.6T + 343T^{2} \) |
| 13 | \( 1 + 0.138T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 6.91T + 5.06e4T^{2} \) |
| 41 | \( 1 + 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 263.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 411.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 1.42T + 2.05e5T^{2} \) |
| 61 | \( 1 + 8.19T + 2.26e5T^{2} \) |
| 67 | \( 1 - 896.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 993.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 299.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 822.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 501.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 774.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 767.T + 9.12e5T^{2} \) |
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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
−9.516276372421905396553790878610, −8.629298159284863852583164065466, −8.136472028827612927864405605776, −7.41968091975114475097985295333, −6.27713573420350868903450832367, −5.10032399064225945993030642388, −3.91938636186883361619444323674, −3.33049441769950087544870769851, −2.13166454640636095197565683431, −0.52804121592186453434079340518,
0.52804121592186453434079340518, 2.13166454640636095197565683431, 3.33049441769950087544870769851, 3.91938636186883361619444323674, 5.10032399064225945993030642388, 6.27713573420350868903450832367, 7.41968091975114475097985295333, 8.136472028827612927864405605776, 8.629298159284863852583164065466, 9.516276372421905396553790878610
