| L(s) = 1 | + 1.61·2-s + 3-s + 0.618·4-s + 1.61·6-s − 2·7-s − 2.23·8-s + 9-s − 3·11-s + 0.618·12-s − 13-s − 3.23·14-s − 4.85·16-s − 4.23·17-s + 1.61·18-s − 6.70·19-s − 2·21-s − 4.85·22-s + 5.38·23-s − 2.23·24-s − 1.61·26-s + 27-s − 1.23·28-s − 3.61·29-s + 8.70·31-s − 3.38·32-s − 3·33-s − 6.85·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.660·6-s − 0.755·7-s − 0.790·8-s + 0.333·9-s − 0.904·11-s + 0.178·12-s − 0.277·13-s − 0.864·14-s − 1.21·16-s − 1.02·17-s + 0.381·18-s − 1.53·19-s − 0.436·21-s − 1.03·22-s + 1.12·23-s − 0.456·24-s − 0.317·26-s + 0.192·27-s − 0.233·28-s − 0.671·29-s + 1.56·31-s − 0.597·32-s − 0.522·33-s − 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 9.14T + 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.709866378813517048946307232499, −8.239439401933478671742089111929, −6.81692100854864692449781172805, −6.57252330089845428474752477339, −5.36004164896512178376169438276, −4.69030035358369051712084731662, −3.83358179241607855263554064043, −2.95888308481646293476430397970, −2.24143243840522682238543152963, 0,
2.24143243840522682238543152963, 2.95888308481646293476430397970, 3.83358179241607855263554064043, 4.69030035358369051712084731662, 5.36004164896512178376169438276, 6.57252330089845428474752477339, 6.81692100854864692449781172805, 8.239439401933478671742089111929, 8.709866378813517048946307232499
