| L(s) = 1 | + 2.67·2-s + 0.123·3-s + 5.14·4-s − 5-s + 0.330·6-s + 7-s + 8.39·8-s − 2.98·9-s − 2.67·10-s − 2.89·11-s + 0.635·12-s − 3.22·13-s + 2.67·14-s − 0.123·15-s + 12.1·16-s − 1.34·17-s − 7.97·18-s − 0.950·19-s − 5.14·20-s + 0.123·21-s − 7.74·22-s − 8.65·23-s + 1.03·24-s + 25-s − 8.63·26-s − 0.739·27-s + 5.14·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 0.0713·3-s + 2.57·4-s − 0.447·5-s + 0.134·6-s + 0.377·7-s + 2.96·8-s − 0.994·9-s − 0.845·10-s − 0.873·11-s + 0.183·12-s − 0.895·13-s + 0.714·14-s − 0.0318·15-s + 3.03·16-s − 0.327·17-s − 1.88·18-s − 0.218·19-s − 1.14·20-s + 0.0269·21-s − 1.65·22-s − 1.80·23-s + 0.211·24-s + 0.200·25-s − 1.69·26-s − 0.142·27-s + 0.971·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 - 0.123T + 3T^{2} \) |
| 11 | \( 1 + 2.89T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 + 1.34T + 17T^{2} \) |
| 19 | \( 1 + 0.950T + 19T^{2} \) |
| 23 | \( 1 + 8.65T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 0.324T + 31T^{2} \) |
| 37 | \( 1 - 0.694T + 37T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 + 3.68T + 43T^{2} \) |
| 47 | \( 1 + 9.76T + 47T^{2} \) |
| 53 | \( 1 + 1.66T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 1.71T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 6.13T + 71T^{2} \) |
| 73 | \( 1 - 5.52T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 + 1.45T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 + 9.88T + 97T^{2} \) |
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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
−7.40858850081539745549142849694, −6.52760853022090402646246766972, −5.86779583143259923349487918029, −5.26942000822646487112539530253, −4.72232733132571387796096637531, −3.96404605491601298356596627912, −3.28284536772822101718338249725, −2.45024434141326349954861087603, −1.95388497812297759352876015813, 0,
1.95388497812297759352876015813, 2.45024434141326349954861087603, 3.28284536772822101718338249725, 3.96404605491601298356596627912, 4.72232733132571387796096637531, 5.26942000822646487112539530253, 5.86779583143259923349487918029, 6.52760853022090402646246766972, 7.40858850081539745549142849694
