| L(s) = 1 | + 1.17·3-s + 5-s − 3.16·7-s − 1.61·9-s − 0.427·11-s + 0.986·13-s + 1.17·15-s + 2.72·17-s − 1.41·19-s − 3.72·21-s + 6.25·23-s + 25-s − 5.42·27-s + 1.45·29-s + 0.966·31-s − 0.501·33-s − 3.16·35-s − 7.21·37-s + 1.15·39-s − 10.4·41-s + 9.12·43-s − 1.61·45-s − 12.8·47-s + 3.04·49-s + 3.20·51-s − 3.99·53-s − 0.427·55-s + ⋯ |
| L(s) = 1 | + 0.678·3-s + 0.447·5-s − 1.19·7-s − 0.539·9-s − 0.128·11-s + 0.273·13-s + 0.303·15-s + 0.660·17-s − 0.325·19-s − 0.812·21-s + 1.30·23-s + 0.200·25-s − 1.04·27-s + 0.269·29-s + 0.173·31-s − 0.0873·33-s − 0.535·35-s − 1.18·37-s + 0.185·39-s − 1.63·41-s + 1.39·43-s − 0.241·45-s − 1.86·47-s + 0.434·49-s + 0.448·51-s − 0.548·53-s − 0.0575·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 + 0.427T + 11T^{2} \) |
| 13 | \( 1 - 0.986T + 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 6.25T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 0.966T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 + 0.170T + 59T^{2} \) |
| 61 | \( 1 + 0.957T + 61T^{2} \) |
| 67 | \( 1 - 1.61T + 67T^{2} \) |
| 71 | \( 1 - 7.83T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 3.09T + 79T^{2} \) |
| 83 | \( 1 - 6.07T + 83T^{2} \) |
| 89 | \( 1 - 5.74T + 89T^{2} \) |
| 97 | \( 1 + 5.12T + 97T^{2} \) |
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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
−7.85743224951617942636007787097, −6.85921520811239511142014830391, −6.44895946332395579698233493277, −5.58936851645823325043570706946, −4.95072668730483479095928028655, −3.70353065174740640158185634393, −3.17270788425848429169067250007, −2.56706738042353185078031330294, −1.42413722760490484930917916240, 0,
1.42413722760490484930917916240, 2.56706738042353185078031330294, 3.17270788425848429169067250007, 3.70353065174740640158185634393, 4.95072668730483479095928028655, 5.58936851645823325043570706946, 6.44895946332395579698233493277, 6.85921520811239511142014830391, 7.85743224951617942636007787097
