| L(s) = 1 | − 5-s − 7-s − 2·13-s + 2·19-s + 25-s + 6·29-s − 8·31-s + 35-s + 4·37-s − 6·41-s + 2·43-s − 6·47-s + 49-s + 6·53-s − 12·59-s − 8·61-s + 2·65-s + 2·67-s + 6·71-s + 2·73-s + 16·79-s − 6·89-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 − 1.43·31-s + 0.169·35-s + 0.657·37-s − 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 1.02·61-s + 0.248·65-s + 0.244·67-s + 0.712·71-s + 0.234·73-s + 1.80·79-s − 0.635·89-s + 0.209·91-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.302950348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302950348\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.72944154389818, −14.98218471283237, −14.78011284311924, −13.89051358223114, −13.60917684759037, −12.80342258765582, −12.31448886759987, −11.96524927584644, −11.16792496072321, −10.77769822216359, −10.03052025221888, −9.536778878279940, −8.987914699678905, −8.269274799877019, −7.720075036284779, −7.137356726879416, −6.563575196796162, −5.884860654758734, −5.114034251577523, −4.608060960531428, −3.767894899086732, −3.190122837814920, −2.461257913590594, −1.519671559269586, −0.4740798824174354,
0.4740798824174354, 1.519671559269586, 2.461257913590594, 3.190122837814920, 3.767894899086732, 4.608060960531428, 5.114034251577523, 5.884860654758734, 6.563575196796162, 7.137356726879416, 7.720075036284779, 8.269274799877019, 8.987914699678905, 9.536778878279940, 10.03052025221888, 10.77769822216359, 11.16792496072321, 11.96524927584644, 12.31448886759987, 12.80342258765582, 13.60917684759037, 13.89051358223114, 14.78011284311924, 14.98218471283237, 15.72944154389818
