| L(s) = 1 | − 2.46·3-s − 4.08·5-s − 4.62·7-s + 3.07·9-s − 4.22·11-s + 4.60·13-s + 10.0·15-s + 0.221·17-s − 0.311·19-s + 11.4·21-s − 7.93·23-s + 11.6·25-s − 0.175·27-s − 4.36·29-s + 3.02·31-s + 10.4·33-s + 18.8·35-s + 7.63·37-s − 11.3·39-s − 8.18·41-s − 4.18·43-s − 12.5·45-s + 2.06·47-s + 14.4·49-s − 0.544·51-s + 10.6·53-s + 17.2·55-s + ⋯ |
| L(s) = 1 | − 1.42·3-s − 1.82·5-s − 1.74·7-s + 1.02·9-s − 1.27·11-s + 1.27·13-s + 2.59·15-s + 0.0536·17-s − 0.0714·19-s + 2.48·21-s − 1.65·23-s + 2.33·25-s − 0.0338·27-s − 0.810·29-s + 0.543·31-s + 1.81·33-s + 3.19·35-s + 1.25·37-s − 1.81·39-s − 1.27·41-s − 0.637·43-s − 1.86·45-s + 0.300·47-s + 2.05·49-s − 0.0762·51-s + 1.46·53-s + 2.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 + 4.08T + 5T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 + 4.22T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 - 0.221T + 17T^{2} \) |
| 19 | \( 1 + 0.311T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 + 4.36T + 29T^{2} \) |
| 31 | \( 1 - 3.02T + 31T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 + 8.18T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 2.06T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 - 8.58T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 + 0.575T + 73T^{2} \) |
| 79 | \( 1 - 3.32T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.115811796659251228875785643230, −7.12568407748401001334074812071, −6.62461401785615937738850954030, −5.87215950628503081550039809248, −5.24459881884588890500236467491, −4.04118973763311064987071850478, −3.74121980990229644564266699228, −2.71450459526975247381138207876, −0.69261247896599395617602347441, 0,
0.69261247896599395617602347441, 2.71450459526975247381138207876, 3.74121980990229644564266699228, 4.04118973763311064987071850478, 5.24459881884588890500236467491, 5.87215950628503081550039809248, 6.62461401785615937738850954030, 7.12568407748401001334074812071, 8.115811796659251228875785643230
