| L(s) = 1 | − 3.25·5-s + 1.95·7-s − 1.86·11-s + 4.83·13-s + 4.17·17-s + 5.05·19-s + 1.26·23-s + 5.62·25-s − 10.5·29-s − 6.26·31-s − 6.38·35-s + 0.409·37-s + 2.65·41-s + 0.317·43-s + 4.08·47-s − 3.16·49-s + 7.15·53-s + 6.08·55-s + 8.72·59-s + 9.89·61-s − 15.7·65-s − 6.55·67-s + 10.0·71-s − 11.8·73-s − 3.65·77-s − 2.70·79-s − 6.18·83-s + ⋯ |
| L(s) = 1 | − 1.45·5-s + 0.740·7-s − 0.562·11-s + 1.34·13-s + 1.01·17-s + 1.15·19-s + 0.262·23-s + 1.12·25-s − 1.95·29-s − 1.12·31-s − 1.07·35-s + 0.0673·37-s + 0.414·41-s + 0.0483·43-s + 0.596·47-s − 0.451·49-s + 0.983·53-s + 0.820·55-s + 1.13·59-s + 1.26·61-s − 1.95·65-s − 0.800·67-s + 1.19·71-s − 1.38·73-s − 0.416·77-s − 0.304·79-s − 0.678·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.617674728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617674728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.25T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 17 | \( 1 - 4.17T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 - 0.409T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 - 0.317T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 - 7.15T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + 6.55T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 2.70T + 79T^{2} \) |
| 83 | \( 1 + 6.18T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.938627982596857963190239578987, −7.54265434306702640783341731814, −6.97051568412638201795509072394, −5.55634857666475256563153293178, −5.45919341499564076399287259331, −4.21630670993666428308396547449, −3.72328019365132016122361540156, −3.04687692289358441661537743787, −1.68610441344605381793480807037, −0.69745094246454166693836091092,
0.69745094246454166693836091092, 1.68610441344605381793480807037, 3.04687692289358441661537743787, 3.72328019365132016122361540156, 4.21630670993666428308396547449, 5.45919341499564076399287259331, 5.55634857666475256563153293178, 6.97051568412638201795509072394, 7.54265434306702640783341731814, 7.938627982596857963190239578987
