L(s) = 1 | − 1.51·2-s + 3.03·3-s + 0.307·4-s + 3.69·5-s − 4.60·6-s − 2.45·7-s + 2.57·8-s + 6.18·9-s − 5.60·10-s + 1.97·11-s + 0.933·12-s + 0.285·13-s + 3.72·14-s + 11.1·15-s − 4.52·16-s + 2.89·17-s − 9.39·18-s + 8.63·19-s + 1.13·20-s − 7.43·21-s − 3.00·22-s + 1.21·23-s + 7.79·24-s + 8.61·25-s − 0.434·26-s + 9.65·27-s − 0.755·28-s + ⋯ |
L(s) = 1 | − 1.07·2-s + 1.74·3-s + 0.153·4-s + 1.65·5-s − 1.87·6-s − 0.927·7-s + 0.908·8-s + 2.06·9-s − 1.77·10-s + 0.595·11-s + 0.269·12-s + 0.0792·13-s + 0.996·14-s + 2.88·15-s − 1.13·16-s + 0.702·17-s − 2.21·18-s + 1.98·19-s + 0.254·20-s − 1.62·21-s − 0.639·22-s + 0.253·23-s + 1.59·24-s + 1.72·25-s − 0.0851·26-s + 1.85·27-s − 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.613137949\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.613137949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2671 | \( 1 - T \) |
good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 3 | \( 1 - 3.03T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 - 1.97T + 11T^{2} \) |
| 13 | \( 1 - 0.285T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 - 8.63T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 2.02T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 - 2.48T + 41T^{2} \) |
| 43 | \( 1 + 8.60T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 0.882T + 53T^{2} \) |
| 59 | \( 1 + 0.471T + 59T^{2} \) |
| 61 | \( 1 - 2.66T + 61T^{2} \) |
| 67 | \( 1 + 2.11T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 3.95T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 3.52T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−9.104381730467014028552557448718, −8.390574809560884295137080565883, −7.46973833229955721364231108436, −6.98930431678973290102760486476, −5.92322037132140657751425184230, −4.95152295297087275409544654990, −3.58950194617927337743581241877, −3.01484576563288553437317781550, −1.88735975834609826280164416126, −1.26592722536952746867650097791,
1.26592722536952746867650097791, 1.88735975834609826280164416126, 3.01484576563288553437317781550, 3.58950194617927337743581241877, 4.95152295297087275409544654990, 5.92322037132140657751425184230, 6.98930431678973290102760486476, 7.46973833229955721364231108436, 8.390574809560884295137080565883, 9.104381730467014028552557448718