| L(s) = 1 | − 3-s + 5·5-s − 26·9-s − 21·11-s + 9·13-s − 5·15-s + 123·17-s − 50·19-s + 180·23-s + 25·25-s + 53·27-s − 197·29-s − 170·31-s + 21·33-s − 80·37-s − 9·39-s − 470·41-s + 270·43-s − 130·45-s − 313·47-s − 123·51-s − 290·53-s − 105·55-s + 50·57-s + 370·59-s − 80·61-s + 45·65-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.447·5-s − 0.962·9-s − 0.575·11-s + 0.192·13-s − 0.0860·15-s + 1.75·17-s − 0.603·19-s + 1.63·23-s + 1/5·25-s + 0.377·27-s − 1.26·29-s − 0.984·31-s + 0.110·33-s − 0.355·37-s − 0.0369·39-s − 1.79·41-s + 0.957·43-s − 0.430·45-s − 0.971·47-s − 0.337·51-s − 0.751·53-s − 0.257·55-s + 0.116·57-s + 0.816·59-s − 0.167·61-s + 0.0858·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 21 T + p^{3} T^{2} \) |
| 13 | \( 1 - 9 T + p^{3} T^{2} \) |
| 17 | \( 1 - 123 T + p^{3} T^{2} \) |
| 19 | \( 1 + 50 T + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 + 197 T + p^{3} T^{2} \) |
| 31 | \( 1 + 170 T + p^{3} T^{2} \) |
| 37 | \( 1 + 80 T + p^{3} T^{2} \) |
| 41 | \( 1 + 470 T + p^{3} T^{2} \) |
| 43 | \( 1 - 270 T + p^{3} T^{2} \) |
| 47 | \( 1 + 313 T + p^{3} T^{2} \) |
| 53 | \( 1 + 290 T + p^{3} T^{2} \) |
| 59 | \( 1 - 370 T + p^{3} T^{2} \) |
| 61 | \( 1 + 80 T + p^{3} T^{2} \) |
| 67 | \( 1 - 470 T + p^{3} T^{2} \) |
| 71 | \( 1 + 712 T + p^{3} T^{2} \) |
| 73 | \( 1 - 330 T + p^{3} T^{2} \) |
| 79 | \( 1 - 457 T + p^{3} T^{2} \) |
| 83 | \( 1 + 820 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1433 T + p^{3} T^{2} \) |
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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
−9.198273896856627730963321742595, −8.415936802781945101494411265351, −7.54900451096121024045492105963, −6.58093734442639821923477649312, −5.49609817468344218517229273479, −5.21425868491893183797790786214, −3.60878773091140251522967783291, −2.76170753631038576269979494094, −1.42991171938148603554474811022, 0,
1.42991171938148603554474811022, 2.76170753631038576269979494094, 3.60878773091140251522967783291, 5.21425868491893183797790786214, 5.49609817468344218517229273479, 6.58093734442639821923477649312, 7.54900451096121024045492105963, 8.415936802781945101494411265351, 9.198273896856627730963321742595
