| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 2·11-s − 15-s + 4·17-s + 19-s + 21-s − 3·23-s + 25-s − 27-s + 2·29-s − 4·31-s − 2·33-s − 35-s + 10·37-s + 7·41-s + 12·43-s + 45-s + 8·47-s − 6·49-s − 4·51-s − 9·53-s + 2·55-s − 57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.258·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.348·33-s − 0.169·35-s + 1.64·37-s + 1.09·41-s + 1.82·43-s + 0.149·45-s + 1.16·47-s − 6/7·49-s − 0.560·51-s − 1.23·53-s + 0.269·55-s − 0.132·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.341947203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.341947203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.63335040007864, −14.16361405797827, −13.97825964773142, −12.96401894751455, −12.76288763677067, −12.22337250600678, −11.69084994204265, −11.01851221247297, −10.73498773117146, −9.900514952555169, −9.552794857494354, −9.209112024523743, −8.343869978703994, −7.680103935464938, −7.287022783712499, −6.474480186523471, −6.000860197084109, −5.695211366422552, −4.894612799729001, −4.230966493803348, −3.671768581724133, −2.865153372231061, −2.165252280205788, −1.243673631564197, −0.6456699745832275,
0.6456699745832275, 1.243673631564197, 2.165252280205788, 2.865153372231061, 3.671768581724133, 4.230966493803348, 4.894612799729001, 5.695211366422552, 6.000860197084109, 6.474480186523471, 7.287022783712499, 7.680103935464938, 8.343869978703994, 9.209112024523743, 9.552794857494354, 9.900514952555169, 10.73498773117146, 11.01851221247297, 11.69084994204265, 12.22337250600678, 12.76288763677067, 12.96401894751455, 13.97825964773142, 14.16361405797827, 14.63335040007864
