| L(s) = 1 | + 2-s − 1.56·3-s + 4-s − 5-s − 1.56·6-s + 8-s − 0.538·9-s − 10-s + 4.17·11-s − 1.56·12-s + 0.720·13-s + 1.56·15-s + 16-s − 17-s − 0.538·18-s − 4.61·19-s − 20-s + 4.17·22-s − 3.09·23-s − 1.56·24-s + 25-s + 0.720·26-s + 5.55·27-s + 5.63·29-s + 1.56·30-s − 4.61·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.905·3-s + 0.5·4-s − 0.447·5-s − 0.640·6-s + 0.353·8-s − 0.179·9-s − 0.316·10-s + 1.25·11-s − 0.452·12-s + 0.199·13-s + 0.405·15-s + 0.250·16-s − 0.242·17-s − 0.126·18-s − 1.05·19-s − 0.223·20-s + 0.890·22-s − 0.644·23-s − 0.320·24-s + 0.200·25-s + 0.141·26-s + 1.06·27-s + 1.04·29-s + 0.286·30-s − 0.829·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.899215450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.899215450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 - 0.720T + 13T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 - 5.39T + 37T^{2} \) |
| 41 | \( 1 - 0.353T + 41T^{2} \) |
| 43 | \( 1 + 6.21T + 43T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 - 5.57T + 53T^{2} \) |
| 59 | \( 1 - 3.95T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 6.08T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 2.47T + 73T^{2} \) |
| 79 | \( 1 - 0.858T + 79T^{2} \) |
| 83 | \( 1 - 2.09T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−7.59433885078642694688622572805, −6.82822130593410304510340554295, −6.26360508551034114394845491322, −5.88165370239789958250247685856, −4.93852979341559890709663217114, −4.31127928911477341479160885044, −3.75263264286587452594824457998, −2.79535334445398416556140151491, −1.75711858086494328396683183198, −0.63829865383793624683062102957,
0.63829865383793624683062102957, 1.75711858086494328396683183198, 2.79535334445398416556140151491, 3.75263264286587452594824457998, 4.31127928911477341479160885044, 4.93852979341559890709663217114, 5.88165370239789958250247685856, 6.26360508551034114394845491322, 6.82822130593410304510340554295, 7.59433885078642694688622572805
