| L(s) = 1 | + 2·7-s − 2·13-s − 3·17-s − 5·19-s − 3·23-s + 6·29-s − 5·31-s − 2·37-s − 12·41-s + 8·43-s + 12·47-s − 3·49-s − 3·53-s + 6·59-s − 7·61-s + 2·67-s + 12·71-s + 16·73-s + 79-s + 15·83-s + 12·89-s − 4·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.554·13-s − 0.727·17-s − 1.14·19-s − 0.625·23-s + 1.11·29-s − 0.898·31-s − 0.328·37-s − 1.87·41-s + 1.21·43-s + 1.75·47-s − 3/7·49-s − 0.412·53-s + 0.781·59-s − 0.896·61-s + 0.244·67-s + 1.42·71-s + 1.87·73-s + 0.112·79-s + 1.64·83-s + 1.27·89-s − 0.419·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.781565959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781565959\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.67130737844207, −15.80910701491507, −15.41271817243633, −14.81090987087447, −14.25238378544721, −13.76867030450861, −13.11459162990338, −12.30831838907869, −12.10123375590090, −11.20969986737207, −10.72714558846332, −10.25900333632812, −9.392382965649691, −8.838537491680785, −8.192274791053258, −7.703853586087188, −6.827444945079294, −6.389312337909401, −5.459039110339767, −4.835209077881277, −4.240825417770379, −3.457543158265949, −2.310428540762721, −1.916685034256464, −0.6036162398492937,
0.6036162398492937, 1.916685034256464, 2.310428540762721, 3.457543158265949, 4.240825417770379, 4.835209077881277, 5.459039110339767, 6.389312337909401, 6.827444945079294, 7.703853586087188, 8.192274791053258, 8.838537491680785, 9.392382965649691, 10.25900333632812, 10.72714558846332, 11.20969986737207, 12.10123375590090, 12.30831838907869, 13.11459162990338, 13.76867030450861, 14.25238378544721, 14.81090987087447, 15.41271817243633, 15.80910701491507, 16.67130737844207
