L(s) = 1 | + 2-s + 3.17·3-s + 4-s − 3.39·5-s + 3.17·6-s + 3.15·7-s + 8-s + 7.04·9-s − 3.39·10-s + 3.33·11-s + 3.17·12-s + 0.449·13-s + 3.15·14-s − 10.7·15-s + 16-s + 6.07·17-s + 7.04·18-s − 5.15·19-s − 3.39·20-s + 9.99·21-s + 3.33·22-s − 0.413·23-s + 3.17·24-s + 6.53·25-s + 0.449·26-s + 12.8·27-s + 3.15·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.83·3-s + 0.5·4-s − 1.51·5-s + 1.29·6-s + 1.19·7-s + 0.353·8-s + 2.34·9-s − 1.07·10-s + 1.00·11-s + 0.915·12-s + 0.124·13-s + 0.843·14-s − 2.77·15-s + 0.250·16-s + 1.47·17-s + 1.66·18-s − 1.18·19-s − 0.759·20-s + 2.18·21-s + 0.711·22-s − 0.0861·23-s + 0.647·24-s + 1.30·25-s + 0.0881·26-s + 2.47·27-s + 0.596·28-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
\(L(1)\) |
\(\approx\) |
\(5.823631748\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.823631748\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 + 3.39T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 - 3.33T + 11T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 19 | \( 1 + 5.15T + 19T^{2} \) |
| 23 | \( 1 + 0.413T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 + 5.86T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 - 6.56T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 + 6.00T + 61T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 3.11T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 8.99T + 97T^{2} \) |
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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
−8.107485475904905155673153849358, −7.81632012156355798689164465576, −7.42302608779196600972819751437, −6.40431481784539815123145870940, −5.12216773153485377969718731476, −4.18849102172704783091022430823, −3.87819324598271074781066365462, −3.27864973447093747652071220073, −2.15060466039136632778812307676, −1.31998874236708161132333099086,
1.31998874236708161132333099086, 2.15060466039136632778812307676, 3.27864973447093747652071220073, 3.87819324598271074781066365462, 4.18849102172704783091022430823, 5.12216773153485377969718731476, 6.40431481784539815123145870940, 7.42302608779196600972819751437, 7.81632012156355798689164465576, 8.107485475904905155673153849358