| L(s) = 1 | + 2.57·3-s − 5-s + 0.619·7-s + 3.61·9-s − 4.29·11-s + 3.48·13-s − 2.57·15-s − 3.08·17-s − 5.05·19-s + 1.59·21-s − 4.10·23-s + 25-s + 1.59·27-s + 6.51·29-s − 0.284·31-s − 11.0·33-s − 0.619·35-s + 4.51·37-s + 8.96·39-s − 6.30·41-s − 3.21·43-s − 3.61·45-s − 1.32·47-s − 6.61·49-s − 7.93·51-s − 0.392·53-s + 4.29·55-s + ⋯ |
| L(s) = 1 | + 1.48·3-s − 0.447·5-s + 0.234·7-s + 1.20·9-s − 1.29·11-s + 0.966·13-s − 0.664·15-s − 0.748·17-s − 1.15·19-s + 0.347·21-s − 0.856·23-s + 0.200·25-s + 0.306·27-s + 1.21·29-s − 0.0510·31-s − 1.92·33-s − 0.104·35-s + 0.741·37-s + 1.43·39-s − 0.984·41-s − 0.490·43-s − 0.539·45-s − 0.193·47-s − 0.945·49-s − 1.11·51-s − 0.0538·53-s + 0.579·55-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 | \( 1 - 2.57T + 3T^{2} \) |
| 7 | \( 1 - 0.619T + 7T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 + 3.08T + 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 + 4.10T + 23T^{2} \) |
| 29 | \( 1 - 6.51T + 29T^{2} \) |
| 31 | \( 1 + 0.284T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 + 6.30T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 + 1.32T + 47T^{2} \) |
| 53 | \( 1 + 0.392T + 53T^{2} \) |
| 59 | \( 1 + 8.91T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 - 4.97T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 9.58T + 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 5.28T + 97T^{2} \) |
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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
−7.910179411184899035003128098887, −6.88706572406435159523604344347, −6.30282718339369180377420311286, −5.25360398478524342446542275589, −4.40528318445789938281739393151, −3.85807539562239856187607546192, −2.98625811984031000289419584470, −2.41134768629659306944145298514, −1.56503940487372944419387662519, 0,
1.56503940487372944419387662519, 2.41134768629659306944145298514, 2.98625811984031000289419584470, 3.85807539562239856187607546192, 4.40528318445789938281739393151, 5.25360398478524342446542275589, 6.30282718339369180377420311286, 6.88706572406435159523604344347, 7.910179411184899035003128098887
