| L(s) = 1 | + 2.60·2-s − 1.83·3-s + 4.78·4-s + 5-s − 4.77·6-s − 0.644·7-s + 7.25·8-s + 0.364·9-s + 2.60·10-s − 11-s − 8.78·12-s − 4.23·13-s − 1.67·14-s − 1.83·15-s + 9.33·16-s − 5.00·17-s + 0.950·18-s − 3.86·19-s + 4.78·20-s + 1.18·21-s − 2.60·22-s − 5.12·23-s − 13.3·24-s + 25-s − 11.0·26-s + 4.83·27-s − 3.08·28-s + ⋯ |
| L(s) = 1 | + 1.84·2-s − 1.05·3-s + 2.39·4-s + 0.447·5-s − 1.95·6-s − 0.243·7-s + 2.56·8-s + 0.121·9-s + 0.823·10-s − 0.301·11-s − 2.53·12-s − 1.17·13-s − 0.448·14-s − 0.473·15-s + 2.33·16-s − 1.21·17-s + 0.224·18-s − 0.885·19-s + 1.07·20-s + 0.258·21-s − 0.555·22-s − 1.06·23-s − 2.71·24-s + 0.200·25-s − 2.16·26-s + 0.930·27-s − 0.583·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 3 | \( 1 + 1.83T + 3T^{2} \) |
| 7 | \( 1 + 0.644T + 7T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 5.00T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 - 0.341T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 - 0.944T + 37T^{2} \) |
| 41 | \( 1 + 6.00T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 + 2.92T + 59T^{2} \) |
| 61 | \( 1 + 6.07T + 61T^{2} \) |
| 67 | \( 1 + 8.36T + 67T^{2} \) |
| 71 | \( 1 - 7.25T + 71T^{2} \) |
| 79 | \( 1 - 5.02T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 2.70T + 89T^{2} \) |
| 97 | \( 1 - 7.05T + 97T^{2} \) |
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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
−7.64855557004629436359392094908, −6.82166246784111223356572038042, −6.29563191580523117721952638128, −5.82063030405945468300798866346, −4.90710922074341023253405791998, −4.67856176477617500752367738671, −3.63390223842079170632326293545, −2.56946504793844877112970187548, −1.96537585026208665706298123545, 0,
1.96537585026208665706298123545, 2.56946504793844877112970187548, 3.63390223842079170632326293545, 4.67856176477617500752367738671, 4.90710922074341023253405791998, 5.82063030405945468300798866346, 6.29563191580523117721952638128, 6.82166246784111223356572038042, 7.64855557004629436359392094908
