| L(s) = 1 | − 13·7-s − 6·11-s + 5·13-s + 78·17-s + 65·19-s − 138·23-s − 66·29-s + 299·31-s − 214·37-s − 360·41-s + 203·43-s − 78·47-s − 174·49-s − 636·53-s − 786·59-s + 467·61-s − 217·67-s + 360·71-s − 286·73-s + 78·77-s + 272·79-s − 498·83-s − 65·91-s − 511·97-s + 1.81e3·101-s − 1.70e3·103-s − 1.23e3·107-s + ⋯ |
| L(s) = 1 | − 0.701·7-s − 0.164·11-s + 0.106·13-s + 1.11·17-s + 0.784·19-s − 1.25·23-s − 0.422·29-s + 1.73·31-s − 0.950·37-s − 1.37·41-s + 0.719·43-s − 0.242·47-s − 0.507·49-s − 1.64·53-s − 1.73·59-s + 0.980·61-s − 0.395·67-s + 0.601·71-s − 0.458·73-s + 0.115·77-s + 0.387·79-s − 0.658·83-s − 0.0748·91-s − 0.534·97-s + 1.78·101-s − 1.63·103-s − 1.11·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 5 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 65 T + p^{3} T^{2} \) |
| 23 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 299 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 360 T + p^{3} T^{2} \) |
| 43 | \( 1 - 203 T + p^{3} T^{2} \) |
| 47 | \( 1 + 78 T + p^{3} T^{2} \) |
| 53 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 786 T + p^{3} T^{2} \) |
| 61 | \( 1 - 467 T + p^{3} T^{2} \) |
| 67 | \( 1 + 217 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 286 T + p^{3} T^{2} \) |
| 79 | \( 1 - 272 T + p^{3} T^{2} \) |
| 83 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 511 T + p^{3} T^{2} \) |
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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
−9.569606235179620032877442203128, −8.353626751179776151582937505479, −7.69917994605502400072877346541, −6.65342552078132698315102450649, −5.87581161801070877355223699777, −4.91970529534829414223163678505, −3.68986477595632227870410713524, −2.88384454312624688425462428218, −1.43026964004570665953934602828, 0,
1.43026964004570665953934602828, 2.88384454312624688425462428218, 3.68986477595632227870410713524, 4.91970529534829414223163678505, 5.87581161801070877355223699777, 6.65342552078132698315102450649, 7.69917994605502400072877346541, 8.353626751179776151582937505479, 9.569606235179620032877442203128
