| L(s) = 1 | + 7-s + 4·11-s − 6·17-s + 6·19-s + 23-s − 5·25-s + 10·29-s + 4·31-s + 2·37-s + 10·41-s + 4·43-s − 12·47-s + 49-s − 6·53-s − 2·59-s + 8·71-s − 6·73-s + 4·77-s − 8·79-s − 14·83-s + 14·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s − 1.45·17-s + 1.37·19-s + 0.208·23-s − 25-s + 1.85·29-s + 0.718·31-s + 0.328·37-s + 1.56·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.260·59-s + 0.949·71-s − 0.702·73-s + 0.455·77-s − 0.900·79-s − 1.53·83-s + 1.48·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.157438252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.157438252\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.82140761379699, −13.51188036888453, −12.82011871213927, −12.31542980896818, −11.78282108253718, −11.27582597544733, −11.21515558778447, −10.26931800344798, −9.856413002573164, −9.289523101445173, −8.953094715363237, −8.267314414822191, −7.892329904717794, −7.180275921473129, −6.727024009802735, −6.179899428093780, −5.773372115839390, −4.817240869971432, −4.556206750534034, −4.011183055867916, −3.228675144594616, −2.694362091703957, −1.917611985875876, −1.253564892661657, −0.6101596329282101,
0.6101596329282101, 1.253564892661657, 1.917611985875876, 2.694362091703957, 3.228675144594616, 4.011183055867916, 4.556206750534034, 4.817240869971432, 5.773372115839390, 6.179899428093780, 6.727024009802735, 7.180275921473129, 7.892329904717794, 8.267314414822191, 8.953094715363237, 9.289523101445173, 9.856413002573164, 10.26931800344798, 11.21515558778447, 11.27582597544733, 11.78282108253718, 12.31542980896818, 12.82011871213927, 13.51188036888453, 13.82140761379699
