| L(s) = 1 | − 2-s − 3.21·3-s + 4-s + 3.26·5-s + 3.21·6-s − 3.74·7-s − 8-s + 7.32·9-s − 3.26·10-s + 2.74·11-s − 3.21·12-s + 1.96·13-s + 3.74·14-s − 10.4·15-s + 16-s + 2.78·17-s − 7.32·18-s + 5.90·19-s + 3.26·20-s + 12.0·21-s − 2.74·22-s + 0.836·23-s + 3.21·24-s + 5.64·25-s − 1.96·26-s − 13.8·27-s − 3.74·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.85·3-s + 0.5·4-s + 1.45·5-s + 1.31·6-s − 1.41·7-s − 0.353·8-s + 2.44·9-s − 1.03·10-s + 0.828·11-s − 0.927·12-s + 0.546·13-s + 1.00·14-s − 2.70·15-s + 0.250·16-s + 0.676·17-s − 1.72·18-s + 1.35·19-s + 0.729·20-s + 2.62·21-s − 0.585·22-s + 0.174·23-s + 0.655·24-s + 1.12·25-s − 0.386·26-s − 2.67·27-s − 0.708·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9964419761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9964419761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 3.21T + 3T^{2} \) |
| 5 | \( 1 - 3.26T + 5T^{2} \) |
| 7 | \( 1 + 3.74T + 7T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 - 1.96T + 13T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 - 0.836T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 - 6.96T + 31T^{2} \) |
| 37 | \( 1 + 9.58T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 6.25T + 43T^{2} \) |
| 47 | \( 1 + 7.60T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 - 5.81T + 61T^{2} \) |
| 67 | \( 1 + 0.257T + 67T^{2} \) |
| 71 | \( 1 + 4.76T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 9.25T + 97T^{2} \) |
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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
−8.726988108927029207728946876352, −7.31180081747430403836086272711, −6.77096912727027219772595065532, −6.15477178730179890262322537968, −5.82686622530483875744798334792, −5.11954862889297337118454923966, −3.88330532786532470467022932802, −2.78523958503143022402121305744, −1.38962416061013637508825601084, −0.799361501230213570513604847577,
0.799361501230213570513604847577, 1.38962416061013637508825601084, 2.78523958503143022402121305744, 3.88330532786532470467022932802, 5.11954862889297337118454923966, 5.82686622530483875744798334792, 6.15477178730179890262322537968, 6.77096912727027219772595065532, 7.31180081747430403836086272711, 8.726988108927029207728946876352
