L(s) = 1 | − 1.11·5-s − 7-s − 5.56·11-s − 3.90·13-s − 2.20·17-s − 19-s − 8.48·23-s − 3.76·25-s − 9.96·29-s − 0.769·31-s + 1.11·35-s − 8.41·37-s + 3.91·41-s + 5.76·43-s − 8.37·47-s + 49-s − 0.170·53-s + 6.19·55-s − 0.887·59-s + 15.0·61-s + 4.34·65-s + 6.78·67-s − 8.76·71-s + 10.4·73-s + 5.56·77-s + 1.31·79-s + 17.7·83-s + ⋯ |
L(s) = 1 | − 0.497·5-s − 0.377·7-s − 1.67·11-s − 1.08·13-s − 0.534·17-s − 0.229·19-s − 1.76·23-s − 0.752·25-s − 1.85·29-s − 0.138·31-s + 0.188·35-s − 1.38·37-s + 0.611·41-s + 0.878·43-s − 1.22·47-s + 0.142·49-s − 0.0234·53-s + 0.835·55-s − 0.115·59-s + 1.93·61-s + 0.538·65-s + 0.829·67-s − 1.03·71-s + 1.22·73-s + 0.634·77-s + 0.147·79-s + 1.94·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1529267269\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1529267269\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 + 3.90T + 13T^{2} \) |
| 17 | \( 1 + 2.20T + 17T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 9.96T + 29T^{2} \) |
| 31 | \( 1 + 0.769T + 31T^{2} \) |
| 37 | \( 1 + 8.41T + 37T^{2} \) |
| 41 | \( 1 - 3.91T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 8.37T + 47T^{2} \) |
| 53 | \( 1 + 0.170T + 53T^{2} \) |
| 59 | \( 1 + 0.887T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 6.78T + 67T^{2} \) |
| 71 | \( 1 + 8.76T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.31T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 9.57T + 89T^{2} \) |
| 97 | \( 1 + 1.04T + 97T^{2} \) |
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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
−7.72805461019370460416940622213, −7.19771219708961826527465257242, −6.32137970828199680028598501722, −5.51392169309094357875171374720, −5.06078166110886297212076056293, −4.08988577814200404330231349244, −3.54398384482339271641226096352, −2.42939921647732905024273719620, −2.04229692128182525463440578991, −0.16991846984722789628292129500,
0.16991846984722789628292129500, 2.04229692128182525463440578991, 2.42939921647732905024273719620, 3.54398384482339271641226096352, 4.08988577814200404330231349244, 5.06078166110886297212076056293, 5.51392169309094357875171374720, 6.32137970828199680028598501722, 7.19771219708961826527465257242, 7.72805461019370460416940622213