| L(s) = 1 | + 0.414·3-s + 5-s − 2.82·9-s + 0.171·11-s + 4.41·13-s + 0.414·15-s + 3.24·17-s − 6·19-s − 7.41·23-s + 25-s − 2.41·27-s − 8.65·29-s − 10.2·31-s + 0.0710·33-s + 2.24·37-s + 1.82·39-s − 6.24·41-s − 2·43-s − 2.82·45-s + 7.24·47-s + 1.34·51-s + 4.24·53-s + 0.171·55-s − 2.48·57-s + 2.24·59-s − 2.82·61-s + 4.41·65-s + ⋯ |
| L(s) = 1 | + 0.239·3-s + 0.447·5-s − 0.942·9-s + 0.0517·11-s + 1.22·13-s + 0.106·15-s + 0.786·17-s − 1.37·19-s − 1.54·23-s + 0.200·25-s − 0.464·27-s − 1.60·29-s − 1.83·31-s + 0.0123·33-s + 0.368·37-s + 0.292·39-s − 0.974·41-s − 0.304·43-s − 0.421·45-s + 1.05·47-s + 0.188·51-s + 0.582·53-s + 0.0231·55-s − 0.329·57-s + 0.291·59-s − 0.362·61-s + 0.547·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 11 | \( 1 - 0.171T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 7.24T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 - 8.24T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 1.48T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.227753749279112102111243207249, −7.48651952874481609730131209524, −6.46731816068341443294093132825, −5.80984011499928232161241832332, −5.39924192322974029671107887965, −3.98357818011455513700172830582, −3.57829852184502176415068054402, −2.37369162524291894141534517724, −1.61885088638283694657200331975, 0,
1.61885088638283694657200331975, 2.37369162524291894141534517724, 3.57829852184502176415068054402, 3.98357818011455513700172830582, 5.39924192322974029671107887965, 5.80984011499928232161241832332, 6.46731816068341443294093132825, 7.48651952874481609730131209524, 8.227753749279112102111243207249
