| L(s) = 1 | − 1.21·2-s + 3-s − 0.525·4-s + 1.31·5-s − 1.21·6-s + 1.52·7-s + 3.06·8-s + 9-s − 1.59·10-s − 11-s − 0.525·12-s − 13-s − 1.85·14-s + 1.31·15-s − 2.67·16-s + 2.21·17-s − 1.21·18-s − 1.52·19-s − 0.688·20-s + 1.52·21-s + 1.21·22-s + 7.95·23-s + 3.06·24-s − 3.28·25-s + 1.21·26-s + 27-s − 0.801·28-s + ⋯ |
| L(s) = 1 | − 0.858·2-s + 0.577·3-s − 0.262·4-s + 0.586·5-s − 0.495·6-s + 0.576·7-s + 1.08·8-s + 0.333·9-s − 0.503·10-s − 0.301·11-s − 0.151·12-s − 0.277·13-s − 0.495·14-s + 0.338·15-s − 0.668·16-s + 0.537·17-s − 0.286·18-s − 0.349·19-s − 0.154·20-s + 0.332·21-s + 0.258·22-s + 1.65·23-s + 0.625·24-s − 0.656·25-s + 0.238·26-s + 0.192·27-s − 0.151·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.158629263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158629263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 17 | \( 1 - 2.21T + 17T^{2} \) |
| 19 | \( 1 + 1.52T + 19T^{2} \) |
| 23 | \( 1 - 7.95T + 23T^{2} \) |
| 29 | \( 1 - 7.39T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 8.77T + 43T^{2} \) |
| 47 | \( 1 - 9.18T + 47T^{2} \) |
| 53 | \( 1 - 3.67T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 5.25T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 3.13T + 73T^{2} \) |
| 79 | \( 1 + 5.03T + 79T^{2} \) |
| 83 | \( 1 - 5.37T + 83T^{2} \) |
| 89 | \( 1 - 0.688T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−10.77730619740489860646333832110, −10.05462720485653575771690319116, −9.351160089592914002948689649139, −8.455259988084925159458662749604, −7.85252243382796363969046546728, −6.78073846958267548280070790431, −5.29347768171698644075866803467, −4.36713674797653644813690419009, −2.71105323986459501041136420495, −1.28290594107518245460998060122,
1.28290594107518245460998060122, 2.71105323986459501041136420495, 4.36713674797653644813690419009, 5.29347768171698644075866803467, 6.78073846958267548280070790431, 7.85252243382796363969046546728, 8.455259988084925159458662749604, 9.351160089592914002948689649139, 10.05462720485653575771690319116, 10.77730619740489860646333832110
