| L(s) = 1 | − 2.25·3-s + 5-s + 0.417·7-s + 2.09·9-s + 2.34·11-s − 4.02·13-s − 2.25·15-s − 0.761·17-s − 4.51·19-s − 0.942·21-s + 0.263·23-s + 25-s + 2.05·27-s + 8.49·29-s + 4.32·31-s − 5.29·33-s + 0.417·35-s − 9.39·37-s + 9.08·39-s + 8.24·41-s + 2.19·43-s + 2.09·45-s − 12.0·47-s − 6.82·49-s + 1.71·51-s + 0.859·53-s + 2.34·55-s + ⋯ |
| L(s) = 1 | − 1.30·3-s + 0.447·5-s + 0.157·7-s + 0.696·9-s + 0.707·11-s − 1.11·13-s − 0.582·15-s − 0.184·17-s − 1.03·19-s − 0.205·21-s + 0.0548·23-s + 0.200·25-s + 0.394·27-s + 1.57·29-s + 0.776·31-s − 0.922·33-s + 0.0705·35-s − 1.54·37-s + 1.45·39-s + 1.28·41-s + 0.334·43-s + 0.311·45-s − 1.76·47-s − 0.975·49-s + 0.240·51-s + 0.118·53-s + 0.316·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 7 | \( 1 - 0.417T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 4.02T + 13T^{2} \) |
| 17 | \( 1 + 0.761T + 17T^{2} \) |
| 19 | \( 1 + 4.51T + 19T^{2} \) |
| 23 | \( 1 - 0.263T + 23T^{2} \) |
| 29 | \( 1 - 8.49T + 29T^{2} \) |
| 31 | \( 1 - 4.32T + 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 0.859T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 7.64T + 61T^{2} \) |
| 67 | \( 1 + 7.08T + 67T^{2} \) |
| 71 | \( 1 - 4.54T + 71T^{2} \) |
| 73 | \( 1 - 6.55T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 5.03T + 83T^{2} \) |
| 89 | \( 1 - 0.565T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.22410196638893480080358127734, −6.50542297030225336580625442904, −6.30607952241067066399272402821, −5.33867438575179633282495278796, −4.80419197397205576476288251503, −4.24577113952640883823491312484, −3.02013278403251516024831036129, −2.11061885192490124770411882811, −1.09242431134733092113569643843, 0,
1.09242431134733092113569643843, 2.11061885192490124770411882811, 3.02013278403251516024831036129, 4.24577113952640883823491312484, 4.80419197397205576476288251503, 5.33867438575179633282495278796, 6.30607952241067066399272402821, 6.50542297030225336580625442904, 7.22410196638893480080358127734
