| L(s) = 1 | − 2.23·5-s − 3.70·11-s − 6.86·13-s + 7.47·17-s + 5.45·19-s − 3.16·23-s + 3.70·29-s − 6.86·31-s − 8.70·37-s − 2.23·41-s − 4.70·43-s + 13.4·47-s − 5.32·53-s + 8.27·55-s − 5.94·59-s − 0.206·61-s + 15.3·65-s − 9.70·67-s − 5.24·71-s + 1.41·73-s + 5·79-s + 8.23·83-s − 16.7·85-s + 11.2·89-s − 12.1·95-s + 0.206·97-s − 3.70·101-s + ⋯ |
| L(s) = 1 | − 0.999·5-s − 1.11·11-s − 1.90·13-s + 1.81·17-s + 1.25·19-s − 0.659·23-s + 0.687·29-s − 1.23·31-s − 1.43·37-s − 0.349·41-s − 0.717·43-s + 1.96·47-s − 0.731·53-s + 1.11·55-s − 0.773·59-s − 0.0264·61-s + 1.90·65-s − 1.18·67-s − 0.622·71-s + 0.165·73-s + 0.562·79-s + 0.904·83-s − 1.81·85-s + 1.19·89-s − 1.25·95-s + 0.0209·97-s − 0.368·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8983025755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8983025755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 + 6.86T + 13T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 5.32T + 53T^{2} \) |
| 59 | \( 1 + 5.94T + 59T^{2} \) |
| 61 | \( 1 + 0.206T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 5.24T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 0.206T + 97T^{2} \) |
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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
−7.83093398320870960031546747829, −7.57318106298963983268376490953, −7.18867189354386065955179553328, −5.83084487871163748197288990054, −5.21006981712603700328024855797, −4.66615767052776025209098814541, −3.51301366877089425764641744713, −3.04096308792951027610143588294, −1.94998605632678707871781663317, −0.49099759992190030560794667285,
0.49099759992190030560794667285, 1.94998605632678707871781663317, 3.04096308792951027610143588294, 3.51301366877089425764641744713, 4.66615767052776025209098814541, 5.21006981712603700328024855797, 5.83084487871163748197288990054, 7.18867189354386065955179553328, 7.57318106298963983268376490953, 7.83093398320870960031546747829
