| L(s) = 1 | − 2.22·2-s + 3-s + 2.92·4-s − 1.63·5-s − 2.22·6-s − 1.90·7-s − 2.06·8-s + 9-s + 3.62·10-s − 4.46·11-s + 2.92·12-s − 1.29·13-s + 4.22·14-s − 1.63·15-s − 1.28·16-s + 17-s − 2.22·18-s − 6.67·19-s − 4.78·20-s − 1.90·21-s + 9.90·22-s + 5.41·23-s − 2.06·24-s − 2.32·25-s + 2.87·26-s + 27-s − 5.57·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.577·3-s + 1.46·4-s − 0.730·5-s − 0.906·6-s − 0.719·7-s − 0.728·8-s + 0.333·9-s + 1.14·10-s − 1.34·11-s + 0.845·12-s − 0.359·13-s + 1.12·14-s − 0.421·15-s − 0.320·16-s + 0.242·17-s − 0.523·18-s − 1.53·19-s − 1.07·20-s − 0.415·21-s + 2.11·22-s + 1.12·23-s − 0.420·24-s − 0.465·25-s + 0.563·26-s + 0.192·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3188760777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3188760777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 19 | \( 1 + 6.67T + 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 + 0.385T + 29T^{2} \) |
| 31 | \( 1 + 4.16T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 + 9.88T + 41T^{2} \) |
| 43 | \( 1 - 2.00T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 9.15T + 71T^{2} \) |
| 73 | \( 1 + 3.90T + 73T^{2} \) |
| 83 | \( 1 - 9.32T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 + 0.111T + 97T^{2} \) |
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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
−8.505640349486320211197160026222, −7.924478502176556930057264280498, −7.19064558278098297134376390566, −6.82601332170379607073018994340, −5.60564662896977345805036946612, −4.58102776858710739880338912117, −3.56344467088249721323820734432, −2.68488769662795557188151768121, −1.87884991867527994529323406409, −0.37678374963793140303504878407,
0.37678374963793140303504878407, 1.87884991867527994529323406409, 2.68488769662795557188151768121, 3.56344467088249721323820734432, 4.58102776858710739880338912117, 5.60564662896977345805036946612, 6.82601332170379607073018994340, 7.19064558278098297134376390566, 7.924478502176556930057264280498, 8.505640349486320211197160026222
