| L(s) = 1 | + 5-s − 7-s + 2·13-s + 2·19-s + 25-s + 6·29-s − 4·31-s − 35-s − 4·37-s − 9·41-s − 43-s + 3·47-s − 6·49-s + 6·53-s − 61-s + 2·65-s − 13·67-s + 12·71-s − 16·73-s − 10·79-s − 12·83-s + 3·89-s − 2·91-s + 2·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.554·13-s + 0.458·19-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.169·35-s − 0.657·37-s − 1.40·41-s − 0.152·43-s + 0.437·47-s − 6/7·49-s + 0.824·53-s − 0.128·61-s + 0.248·65-s − 1.58·67-s + 1.42·71-s − 1.87·73-s − 1.12·79-s − 1.31·83-s + 0.317·89-s − 0.209·91-s + 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.93545586618011, −15.29683733110204, −14.72827447992851, −14.11717160686752, −13.61660674242144, −13.21981679641475, −12.59987414196933, −11.97822112200753, −11.54441913803194, −10.74196402514580, −10.32646209949514, −9.774626558755615, −9.173615882861238, −8.568176726358413, −8.130955777689162, −7.128475757041908, −6.878700948915784, −6.038575944326981, −5.604046512703937, −4.876849429568776, −4.169314453510117, −3.338993738383419, −2.861196409852831, −1.862365947715089, −1.176527112536176, 0,
1.176527112536176, 1.862365947715089, 2.861196409852831, 3.338993738383419, 4.169314453510117, 4.876849429568776, 5.604046512703937, 6.038575944326981, 6.878700948915784, 7.128475757041908, 8.130955777689162, 8.568176726358413, 9.173615882861238, 9.774626558755615, 10.32646209949514, 10.74196402514580, 11.54441913803194, 11.97822112200753, 12.59987414196933, 13.21981679641475, 13.61660674242144, 14.11717160686752, 14.72827447992851, 15.29683733110204, 15.93545586618011
