L(s) = 1 | − 2-s − 1.27·3-s + 4-s − 0.526·5-s + 1.27·6-s − 8-s − 1.38·9-s + 0.526·10-s − 5.51·11-s − 1.27·12-s + 1.63·13-s + 0.669·15-s + 16-s + 17-s + 1.38·18-s − 7.01·19-s − 0.526·20-s + 5.51·22-s + 8.62·23-s + 1.27·24-s − 4.72·25-s − 1.63·26-s + 5.57·27-s − 5.45·29-s − 0.669·30-s + 2.74·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.734·3-s + 0.5·4-s − 0.235·5-s + 0.519·6-s − 0.353·8-s − 0.461·9-s + 0.166·10-s − 1.66·11-s − 0.367·12-s + 0.454·13-s + 0.172·15-s + 0.250·16-s + 0.242·17-s + 0.326·18-s − 1.60·19-s − 0.117·20-s + 1.17·22-s + 1.79·23-s + 0.259·24-s − 0.944·25-s − 0.321·26-s + 1.07·27-s − 1.01·29-s − 0.122·30-s + 0.492·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5070455749\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5070455749\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 + 0.526T + 5T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 19 | \( 1 + 7.01T + 19T^{2} \) |
| 23 | \( 1 - 8.62T + 23T^{2} \) |
| 29 | \( 1 + 5.45T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 + 9.02T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 4.60T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 - 2.03T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 1.81T + 89T^{2} \) |
| 97 | \( 1 + 3.69T + 97T^{2} \) |
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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
−9.238115840909194681695362989658, −8.590753985587920722502798732217, −7.81564400846598571528195597193, −7.10644401879373201392241967031, −6.04614697122312752378043334082, −5.51729988169660125762339825034, −4.51718876498345516101088255345, −3.16897423999265717763804753927, −2.19532006610552308380468082805, −0.54084090422646518573898400690,
0.54084090422646518573898400690, 2.19532006610552308380468082805, 3.16897423999265717763804753927, 4.51718876498345516101088255345, 5.51729988169660125762339825034, 6.04614697122312752378043334082, 7.10644401879373201392241967031, 7.81564400846598571528195597193, 8.590753985587920722502798732217, 9.238115840909194681695362989658