| L(s) = 1 | − 5-s − 11-s − 2·13-s − 3·17-s + 7·19-s + 3·23-s + 25-s + 3·29-s + 4·31-s − 10·37-s − 7·43-s − 9·53-s + 55-s + 3·59-s + 61-s + 2·65-s + 2·67-s + 12·71-s + 4·73-s + 2·79-s − 3·83-s + 3·85-s + 9·89-s − 7·95-s + 19·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 0.727·17-s + 1.60·19-s + 0.625·23-s + 1/5·25-s + 0.557·29-s + 0.718·31-s − 1.64·37-s − 1.06·43-s − 1.23·53-s + 0.134·55-s + 0.390·59-s + 0.128·61-s + 0.248·65-s + 0.244·67-s + 1.42·71-s + 0.468·73-s + 0.225·79-s − 0.329·83-s + 0.325·85-s + 0.953·89-s − 0.718·95-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637164879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637164879\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−13.79825968286359, −13.30528893872252, −12.84264648359789, −12.15489236463027, −11.91654396190016, −11.39818015754927, −10.85333085468666, −10.35827733736548, −9.799712560595543, −9.388514974711134, −8.734161925430276, −8.326388755979009, −7.663825223152323, −7.348238966111674, −6.622028497016732, −6.397683283155177, −5.253188666088501, −5.157743075097490, −4.601816730669488, −3.763545565355545, −3.276016508626051, −2.727256344920190, −2.003459483946154, −1.196980711325670, −0.4310786534582770,
0.4310786534582770, 1.196980711325670, 2.003459483946154, 2.727256344920190, 3.276016508626051, 3.763545565355545, 4.601816730669488, 5.157743075097490, 5.253188666088501, 6.397683283155177, 6.622028497016732, 7.348238966111674, 7.663825223152323, 8.326388755979009, 8.734161925430276, 9.388514974711134, 9.799712560595543, 10.35827733736548, 10.85333085468666, 11.39818015754927, 11.91654396190016, 12.15489236463027, 12.84264648359789, 13.30528893872252, 13.79825968286359
