| L(s) = 1 | − 5-s − 7-s − 3·9-s − 5·13-s + 17-s + 6·19-s − 3·23-s − 4·25-s − 3·29-s + 2·31-s + 35-s − 2·37-s − 2·41-s − 43-s + 3·45-s + 8·47-s + 49-s − 9·53-s − 59-s − 10·61-s + 3·63-s + 5·65-s − 8·67-s + 11·71-s − 2·73-s − 4·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s − 1.38·13-s + 0.242·17-s + 1.37·19-s − 0.625·23-s − 4/5·25-s − 0.557·29-s + 0.359·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s − 0.152·43-s + 0.447·45-s + 1.16·47-s + 1/7·49-s − 1.23·53-s − 0.130·59-s − 1.28·61-s + 0.377·63-s + 0.620·65-s − 0.977·67-s + 1.30·71-s − 0.234·73-s − 0.450·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.24716471585223, −12.37210232648513, −12.26665457661780, −11.84489837286329, −11.42676244327546, −10.89183659202505, −10.32875939355518, −9.851290391301850, −9.339218521062095, −9.179094153180881, −8.327080183848651, −7.883638110155884, −7.606483035504758, −7.055037212994021, −6.523069686975324, −5.792341149639718, −5.563998908150662, −4.983384210008053, −4.403891324257826, −3.808397678908627, −3.160948236478793, −2.883863151002959, −2.186469432883732, −1.525896506773759, −0.5560668314170061, 0,
0.5560668314170061, 1.525896506773759, 2.186469432883732, 2.883863151002959, 3.160948236478793, 3.808397678908627, 4.403891324257826, 4.983384210008053, 5.563998908150662, 5.792341149639718, 6.523069686975324, 7.055037212994021, 7.606483035504758, 7.883638110155884, 8.327080183848651, 9.179094153180881, 9.339218521062095, 9.851290391301850, 10.32875939355518, 10.89183659202505, 11.42676244327546, 11.84489837286329, 12.26665457661780, 12.37210232648513, 13.24716471585223
