| L(s) = 1 | − 5·5-s + 16·7-s − 28·11-s − 26·13-s + 62·17-s + 68·19-s − 208·23-s + 25·25-s + 58·29-s − 160·31-s − 80·35-s + 270·37-s − 282·41-s − 76·43-s − 280·47-s − 87·49-s + 210·53-s + 140·55-s + 196·59-s + 742·61-s + 130·65-s − 836·67-s − 504·71-s − 1.06e3·73-s − 448·77-s − 768·79-s − 1.05e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.863·7-s − 0.767·11-s − 0.554·13-s + 0.884·17-s + 0.821·19-s − 1.88·23-s + 1/5·25-s + 0.371·29-s − 0.926·31-s − 0.386·35-s + 1.19·37-s − 1.07·41-s − 0.269·43-s − 0.868·47-s − 0.253·49-s + 0.544·53-s + 0.343·55-s + 0.432·59-s + 1.55·61-s + 0.248·65-s − 1.52·67-s − 0.842·71-s − 1.70·73-s − 0.663·77-s − 1.09·79-s − 1.39·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| good | 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 62 T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 + 208 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 270 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 210 T + p^{3} T^{2} \) |
| 59 | \( 1 - 196 T + p^{3} T^{2} \) |
| 61 | \( 1 - 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 836 T + p^{3} T^{2} \) |
| 71 | \( 1 + 504 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1062 T + p^{3} T^{2} \) |
| 79 | \( 1 + 768 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 - 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1406 T + p^{3} T^{2} \) |
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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
−9.800095994046997545597161350359, −8.478744813245837004359988632783, −7.86523313374082870419126362774, −7.22828625858627094936085097158, −5.82590851836972082420520483793, −5.05455036767600041321342701724, −4.05367443821206399974325006276, −2.84483562175671081789187281342, −1.55331718631072820004742209192, 0,
1.55331718631072820004742209192, 2.84483562175671081789187281342, 4.05367443821206399974325006276, 5.05455036767600041321342701724, 5.82590851836972082420520483793, 7.22828625858627094936085097158, 7.86523313374082870419126362774, 8.478744813245837004359988632783, 9.800095994046997545597161350359
