L(s) = 1 | − 2.10·2-s + 3-s + 2.41·4-s − 2.56·5-s − 2.10·6-s − 4.38·7-s − 0.862·8-s + 9-s + 5.37·10-s − 4.74·11-s + 2.41·12-s + 13-s + 9.21·14-s − 2.56·15-s − 3.01·16-s + 5.66·17-s − 2.10·18-s + 2.69·19-s − 6.17·20-s − 4.38·21-s + 9.96·22-s + 2.40·23-s − 0.862·24-s + 1.56·25-s − 2.10·26-s + 27-s − 10.5·28-s + ⋯ |
L(s) = 1 | − 1.48·2-s + 0.577·3-s + 1.20·4-s − 1.14·5-s − 0.857·6-s − 1.65·7-s − 0.304·8-s + 0.333·9-s + 1.70·10-s − 1.43·11-s + 0.695·12-s + 0.277·13-s + 2.46·14-s − 0.661·15-s − 0.752·16-s + 1.37·17-s − 0.495·18-s + 0.618·19-s − 1.38·20-s − 0.957·21-s + 2.12·22-s + 0.501·23-s − 0.175·24-s + 0.312·25-s − 0.411·26-s + 0.192·27-s − 1.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 + 4.38T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 - 3.61T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 - 0.321T + 43T^{2} \) |
| 47 | \( 1 - 6.05T + 47T^{2} \) |
| 53 | \( 1 - 4.84T + 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 + 2.72T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 - 3.53T + 89T^{2} \) |
| 97 | \( 1 - 8.12T + 97T^{2} \) |
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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
−8.016966986058216035959827100597, −7.56527980956031557898272896716, −7.17676449131487810345523764176, −6.15005385907356058126642317113, −5.14505862261247205999112854521, −3.88072274809835171105636555333, −3.21498331903324496894460271547, −2.50835939092684474157744049666, −0.954770687595189254061385045420, 0,
0.954770687595189254061385045420, 2.50835939092684474157744049666, 3.21498331903324496894460271547, 3.88072274809835171105636555333, 5.14505862261247205999112854521, 6.15005385907356058126642317113, 7.17676449131487810345523764176, 7.56527980956031557898272896716, 8.016966986058216035959827100597