L(s) = 1 | − 1.82·2-s − 3-s + 1.33·4-s + 0.0479·5-s + 1.82·6-s − 0.416·7-s + 1.21·8-s + 9-s − 0.0876·10-s + 2.02·11-s − 1.33·12-s − 4.18·13-s + 0.761·14-s − 0.0479·15-s − 4.88·16-s + 17-s − 1.82·18-s − 3.54·19-s + 0.0640·20-s + 0.416·21-s − 3.68·22-s − 4.19·23-s − 1.21·24-s − 4.99·25-s + 7.64·26-s − 27-s − 0.556·28-s + ⋯ |
L(s) = 1 | − 1.29·2-s − 0.577·3-s + 0.667·4-s + 0.0214·5-s + 0.745·6-s − 0.157·7-s + 0.428·8-s + 0.333·9-s − 0.0277·10-s + 0.609·11-s − 0.385·12-s − 1.16·13-s + 0.203·14-s − 0.0123·15-s − 1.22·16-s + 0.242·17-s − 0.430·18-s − 0.813·19-s + 0.0143·20-s + 0.0909·21-s − 0.786·22-s − 0.875·23-s − 0.247·24-s − 0.999·25-s + 1.49·26-s − 0.192·27-s − 0.105·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2731291503\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2731291503\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 5 | \( 1 - 0.0479T + 5T^{2} \) |
| 7 | \( 1 + 0.416T + 7T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 13 | \( 1 + 4.18T + 13T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 0.335T + 29T^{2} \) |
| 31 | \( 1 + 8.59T + 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 + 4.65T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 6.64T + 67T^{2} \) |
| 71 | \( 1 - 2.46T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 3.38T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.980863236051888223077092451341, −7.19627683631495267238209721856, −6.70536438471146216180501248001, −5.92721389140914434389131114994, −5.06976386253629181257227196664, −4.35074864939430520435546064596, −3.53981018588727803455381096396, −2.18062967294961392015343480125, −1.61938900759532371269300642423, −0.32459617194791684929793950288,
0.32459617194791684929793950288, 1.61938900759532371269300642423, 2.18062967294961392015343480125, 3.53981018588727803455381096396, 4.35074864939430520435546064596, 5.06976386253629181257227196664, 5.92721389140914434389131114994, 6.70536438471146216180501248001, 7.19627683631495267238209721856, 7.980863236051888223077092451341