| L(s) = 1 | − 0.381·2-s − 0.970·3-s − 1.85·4-s + 5-s + 0.370·6-s − 3.53·7-s + 1.47·8-s − 2.05·9-s − 0.381·10-s − 5.21·11-s + 1.80·12-s − 1.26·13-s + 1.34·14-s − 0.970·15-s + 3.14·16-s − 17-s + 0.785·18-s + 3.34·19-s − 1.85·20-s + 3.43·21-s + 1.99·22-s + 2.34·23-s − 1.42·24-s + 25-s + 0.484·26-s + 4.90·27-s + 6.55·28-s + ⋯ |
| L(s) = 1 | − 0.269·2-s − 0.560·3-s − 0.927·4-s + 0.447·5-s + 0.151·6-s − 1.33·7-s + 0.520·8-s − 0.685·9-s − 0.120·10-s − 1.57·11-s + 0.519·12-s − 0.351·13-s + 0.360·14-s − 0.250·15-s + 0.786·16-s − 0.242·17-s + 0.185·18-s + 0.767·19-s − 0.414·20-s + 0.748·21-s + 0.424·22-s + 0.489·23-s − 0.291·24-s + 0.200·25-s + 0.0949·26-s + 0.944·27-s + 1.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 3 | \( 1 + 0.970T + 3T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 - 6.91T + 41T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 2.25T + 53T^{2} \) |
| 59 | \( 1 + 0.307T + 59T^{2} \) |
| 61 | \( 1 - 8.30T + 61T^{2} \) |
| 67 | \( 1 - 0.254T + 67T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 0.689T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 3.50T + 89T^{2} \) |
| 97 | \( 1 + 3.29T + 97T^{2} \) |
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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
−7.72433142812394629829426828711, −7.06864399009184393674328889509, −6.09152951301943204947568868745, −5.56356200403571644259792214950, −5.06233775891760581531816447205, −4.14140314646879345350096778251, −3.02846384972186506772618760194, −2.56791615374484140664748348240, −0.842665482652842872710761440626, 0,
0.842665482652842872710761440626, 2.56791615374484140664748348240, 3.02846384972186506772618760194, 4.14140314646879345350096778251, 5.06233775891760581531816447205, 5.56356200403571644259792214950, 6.09152951301943204947568868745, 7.06864399009184393674328889509, 7.72433142812394629829426828711
