| L(s) = 1 | − 2.73·3-s + 5-s + 1.81·7-s + 4.46·9-s − 4.80·11-s − 3.54·13-s − 2.73·15-s − 1.11·17-s + 5.63·19-s − 4.96·21-s + 4.39·23-s + 25-s − 4.01·27-s − 0.434·29-s − 7.95·31-s + 13.1·33-s + 1.81·35-s − 3.14·37-s + 9.68·39-s − 1.77·41-s + 6.23·43-s + 4.46·45-s + 4.42·47-s − 3.70·49-s + 3.05·51-s + 4.03·53-s − 4.80·55-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 0.447·5-s + 0.686·7-s + 1.48·9-s − 1.44·11-s − 0.982·13-s − 0.705·15-s − 0.271·17-s + 1.29·19-s − 1.08·21-s + 0.917·23-s + 0.200·25-s − 0.772·27-s − 0.0807·29-s − 1.42·31-s + 2.28·33-s + 0.306·35-s − 0.516·37-s + 1.55·39-s − 0.277·41-s + 0.950·43-s + 0.666·45-s + 0.645·47-s − 0.529·49-s + 0.428·51-s + 0.554·53-s − 0.647·55-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 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 1.81T + 7T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 + 0.434T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 + 3.14T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 4.42T + 47T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 1.21T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 6.23T + 73T^{2} \) |
| 79 | \( 1 - 6.17T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−7.19716097153752781723747312106, −6.97235688527901363087876739860, −5.74412481231211520493127905423, −5.34632921295641485383415906606, −5.09265447073998142192791430972, −4.24414736937846267695047750845, −2.97654396920467509172269048732, −2.10877526788500416395592819132, −1.04364258929196737061086736485, 0,
1.04364258929196737061086736485, 2.10877526788500416395592819132, 2.97654396920467509172269048732, 4.24414736937846267695047750845, 5.09265447073998142192791430972, 5.34632921295641485383415906606, 5.74412481231211520493127905423, 6.97235688527901363087876739860, 7.19716097153752781723747312106
