| L(s) = 1 | − 2.82·7-s − 5.65·11-s + 2·13-s + 2·17-s − 2.82·23-s − 6·29-s − 5.65·31-s + 10·37-s − 2·41-s − 8.48·43-s + 2.82·47-s + 1.00·49-s + 6·53-s + 11.3·59-s − 2·61-s − 2.82·67-s + 5.65·71-s + 6·73-s + 16.0·77-s − 11.3·79-s − 2.82·83-s − 10·89-s − 5.65·91-s − 2·97-s + 2·101-s + 14.1·103-s + 14.1·107-s + ⋯ |
| L(s) = 1 | − 1.06·7-s − 1.70·11-s + 0.554·13-s + 0.485·17-s − 0.589·23-s − 1.11·29-s − 1.01·31-s + 1.64·37-s − 0.312·41-s − 1.29·43-s + 0.412·47-s + 0.142·49-s + 0.824·53-s + 1.47·59-s − 0.256·61-s − 0.345·67-s + 0.671·71-s + 0.702·73-s + 1.82·77-s − 1.27·79-s − 0.310·83-s − 1.05·89-s − 0.592·91-s − 0.203·97-s + 0.199·101-s + 1.39·103-s + 1.36·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003359643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003359643\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.888741344818032251001908181943, −7.29032545485371000584885007411, −6.51712836887501865120675784078, −5.67951365770390818258446695246, −5.37717874735943276420743381115, −4.24128894933764796336293531791, −3.46676984338734785644367617634, −2.80438642072844496316530101248, −1.91656597738313881477065527347, −0.48183596480629773473459067339,
0.48183596480629773473459067339, 1.91656597738313881477065527347, 2.80438642072844496316530101248, 3.46676984338734785644367617634, 4.24128894933764796336293531791, 5.37717874735943276420743381115, 5.67951365770390818258446695246, 6.51712836887501865120675784078, 7.29032545485371000584885007411, 7.888741344818032251001908181943
