| L(s) = 1 | − 3.52·2-s + 4.39·4-s − 9.17·5-s + 12.6·8-s + 32.2·10-s + 20.4·11-s − 79.8·16-s − 40.3·20-s − 72.0·22-s − 40.8·25-s + 179.·32-s − 116.·40-s − 196.·41-s − 452·43-s + 89.9·44-s + 640.·47-s − 343·49-s + 143.·50-s − 187.·55-s + 579.·59-s + 944.·61-s + 6.63·64-s + 1.19e3·71-s − 418.·79-s + 732.·80-s + 693.·82-s + 94.6·83-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.549·4-s − 0.820·5-s + 0.560·8-s + 1.02·10-s + 0.561·11-s − 1.24·16-s − 0.450·20-s − 0.698·22-s − 0.326·25-s + 0.991·32-s − 0.460·40-s − 0.750·41-s − 1.60·43-s + 0.308·44-s + 1.98·47-s − 49-s + 0.406·50-s − 0.460·55-s + 1.27·59-s + 1.98·61-s + 0.0129·64-s + 1.98·71-s − 0.595·79-s + 1.02·80-s + 0.933·82-s + 0.125·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.52T + 8T^{2} \) |
| 5 | \( 1 + 9.17T + 125T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 - 20.4T + 1.33e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 + 196.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 452T + 7.95e4T^{2} \) |
| 47 | \( 1 - 640.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 - 579.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 944.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.19e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 + 418.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 94.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.67e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−8.483897910118657352071753898326, −8.247429746656681197432700436546, −7.23495663221123530047904824186, −6.71455830302494320646054356260, −5.40733163364397188505797768215, −4.34302551539624240450587668861, −3.56493847822622501583319031591, −2.13409999330796236444640788122, −1.00429582563569572043564346386, 0,
1.00429582563569572043564346386, 2.13409999330796236444640788122, 3.56493847822622501583319031591, 4.34302551539624240450587668861, 5.40733163364397188505797768215, 6.71455830302494320646054356260, 7.23495663221123530047904824186, 8.247429746656681197432700436546, 8.483897910118657352071753898326
