| L(s) = 1 | − 2-s + 3-s + 4-s + 3.41·5-s − 6-s − 4.87·7-s − 8-s + 9-s − 3.41·10-s − 3.41·11-s + 12-s + 2.71·13-s + 4.87·14-s + 3.41·15-s + 16-s + 1.18·17-s − 18-s + 3.41·20-s − 4.87·21-s + 3.41·22-s − 3.41·23-s − 24-s + 6.63·25-s − 2.71·26-s + 27-s − 4.87·28-s + 3.77·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.52·5-s − 0.408·6-s − 1.84·7-s − 0.353·8-s + 0.333·9-s − 1.07·10-s − 1.02·11-s + 0.288·12-s + 0.753·13-s + 1.30·14-s + 0.880·15-s + 0.250·16-s + 0.287·17-s − 0.235·18-s + 0.762·20-s − 1.06·21-s + 0.727·22-s − 0.711·23-s − 0.204·24-s + 1.32·25-s − 0.532·26-s + 0.192·27-s − 0.922·28-s + 0.700·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680695389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680695389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 1.18T + 17T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 - 6.75T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 - 2.85T + 47T^{2} \) |
| 53 | \( 1 - 0.630T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 6.90T + 61T^{2} \) |
| 67 | \( 1 - 0.709T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 0.297T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 6.77T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−9.246151845558999117173930819259, −8.492121808637958123472389198049, −7.62994267359721453776969219689, −6.57735867569920891938163091677, −6.17659808594207035067857375955, −5.44218301792740047394541106532, −3.91414251119240289922174008415, −2.76392115996403388022789798121, −2.42776452325935218149041430937, −0.915812584207097809578931467774,
0.915812584207097809578931467774, 2.42776452325935218149041430937, 2.76392115996403388022789798121, 3.91414251119240289922174008415, 5.44218301792740047394541106532, 6.17659808594207035067857375955, 6.57735867569920891938163091677, 7.62994267359721453776969219689, 8.492121808637958123472389198049, 9.246151845558999117173930819259
