| L(s) = 1 | − 2-s + 3.17·3-s + 4-s − 0.981·5-s − 3.17·6-s − 4.36·7-s − 8-s + 7.06·9-s + 0.981·10-s − 5.66·11-s + 3.17·12-s + 4.51·13-s + 4.36·14-s − 3.11·15-s + 16-s + 17-s − 7.06·18-s + 6.02·19-s − 0.981·20-s − 13.8·21-s + 5.66·22-s + 8.45·23-s − 3.17·24-s − 4.03·25-s − 4.51·26-s + 12.8·27-s − 4.36·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.83·3-s + 0.5·4-s − 0.438·5-s − 1.29·6-s − 1.65·7-s − 0.353·8-s + 2.35·9-s + 0.310·10-s − 1.70·11-s + 0.915·12-s + 1.25·13-s + 1.16·14-s − 0.804·15-s + 0.250·16-s + 0.242·17-s − 1.66·18-s + 1.38·19-s − 0.219·20-s − 3.02·21-s + 1.20·22-s + 1.76·23-s − 0.647·24-s − 0.807·25-s − 0.884·26-s + 2.48·27-s − 0.825·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.958813203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958813203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 + 0.981T + 5T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 11 | \( 1 + 5.66T + 11T^{2} \) |
| 13 | \( 1 - 4.51T + 13T^{2} \) |
| 19 | \( 1 - 6.02T + 19T^{2} \) |
| 23 | \( 1 - 8.45T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 - 7.63T + 31T^{2} \) |
| 37 | \( 1 + 1.22T + 37T^{2} \) |
| 41 | \( 1 + 1.81T + 41T^{2} \) |
| 43 | \( 1 - 0.435T + 43T^{2} \) |
| 47 | \( 1 + 5.55T + 47T^{2} \) |
| 53 | \( 1 + 5.88T + 53T^{2} \) |
| 61 | \( 1 - 5.82T + 61T^{2} \) |
| 67 | \( 1 - 1.75T + 67T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 - 8.01T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 + 2.56T + 97T^{2} \) |
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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
−9.138034218016643011545386288853, −8.289231840209504136640559360749, −7.937623531165673393569024194037, −7.09831400269449964443938220972, −6.39641540960465299507425728021, −5.05411337280394396202005377960, −3.60344104574646633516617528059, −3.12940831769674819868333592551, −2.57219291392870637320105223320, −0.963111818948042605463566793656,
0.963111818948042605463566793656, 2.57219291392870637320105223320, 3.12940831769674819868333592551, 3.60344104574646633516617528059, 5.05411337280394396202005377960, 6.39641540960465299507425728021, 7.09831400269449964443938220972, 7.937623531165673393569024194037, 8.289231840209504136640559360749, 9.138034218016643011545386288853
