L(s) = 1 | + 2-s + 0.381·3-s + 4-s − 5-s + 0.381·6-s + 7-s + 8-s − 2.85·9-s − 10-s + 0.381·12-s + 2·13-s + 14-s − 0.381·15-s + 16-s + 0.618·17-s − 2.85·18-s + 0.145·19-s − 20-s + 0.381·21-s + 6·23-s + 0.381·24-s + 25-s + 2·26-s − 2.23·27-s + 28-s − 1.23·29-s − 0.381·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.220·3-s + 0.5·4-s − 0.447·5-s + 0.155·6-s + 0.377·7-s + 0.353·8-s − 0.951·9-s − 0.316·10-s + 0.110·12-s + 0.554·13-s + 0.267·14-s − 0.0986·15-s + 0.250·16-s + 0.149·17-s − 0.672·18-s + 0.0334·19-s − 0.223·20-s + 0.0833·21-s + 1.25·23-s + 0.0779·24-s + 0.200·25-s + 0.392·26-s − 0.430·27-s + 0.188·28-s − 0.229·29-s − 0.0697·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.332926674\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.332926674\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 - 0.145T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 5.32T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + 3.23T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 - 3.70T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 - 0.472T + 79T^{2} \) |
| 83 | \( 1 + 5.32T + 83T^{2} \) |
| 89 | \( 1 - 1.90T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.74375263481553139393716318327, −7.12516277381347191447011872038, −6.23974685029966967146232500651, −5.73201778475460812888489047479, −4.89580090978530951344422081756, −4.33915818886218504143875291355, −3.34961031769658516483121276259, −2.95760442444676287034331369049, −1.93047527555627234158601924663, −0.796403716217462582161886943804,
0.796403716217462582161886943804, 1.93047527555627234158601924663, 2.95760442444676287034331369049, 3.34961031769658516483121276259, 4.33915818886218504143875291355, 4.89580090978530951344422081756, 5.73201778475460812888489047479, 6.23974685029966967146232500651, 7.12516277381347191447011872038, 7.74375263481553139393716318327