L(s) = 1 | − 2-s − 0.452·3-s + 4-s + 3.47·5-s + 0.452·6-s + 1.39·7-s − 8-s − 2.79·9-s − 3.47·10-s − 0.452·12-s + 2.25·13-s − 1.39·14-s − 1.57·15-s + 16-s − 6.80·17-s + 2.79·18-s + 2.65·19-s + 3.47·20-s − 0.631·21-s − 23-s + 0.452·24-s + 7.08·25-s − 2.25·26-s + 2.62·27-s + 1.39·28-s − 5.90·29-s + 1.57·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.261·3-s + 0.5·4-s + 1.55·5-s + 0.184·6-s + 0.526·7-s − 0.353·8-s − 0.931·9-s − 1.09·10-s − 0.130·12-s + 0.625·13-s − 0.372·14-s − 0.406·15-s + 0.250·16-s − 1.64·17-s + 0.658·18-s + 0.608·19-s + 0.777·20-s − 0.137·21-s − 0.208·23-s + 0.0924·24-s + 1.41·25-s − 0.442·26-s + 0.505·27-s + 0.263·28-s − 1.09·29-s + 0.287·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.691024567\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.691024567\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 0.452T + 3T^{2} \) |
| 5 | \( 1 - 3.47T + 5T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 + 6.80T + 17T^{2} \) |
| 19 | \( 1 - 2.65T + 19T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 4.80T + 61T^{2} \) |
| 67 | \( 1 + 4.52T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 3.44T + 79T^{2} \) |
| 83 | \( 1 + 1.38T + 83T^{2} \) |
| 89 | \( 1 - 9.56T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.435593919882535579859688276586, −7.47267751555080687676226229744, −6.47144653747656557347241339537, −6.21631277284853919170115160497, −5.39883800974650811023086667670, −4.78168724088652532060316861656, −3.47809621286181252291744538215, −2.38486356809249407204595818926, −1.93128087171225216110827814990, −0.78120502925255865566729525402,
0.78120502925255865566729525402, 1.93128087171225216110827814990, 2.38486356809249407204595818926, 3.47809621286181252291744538215, 4.78168724088652532060316861656, 5.39883800974650811023086667670, 6.21631277284853919170115160497, 6.47144653747656557347241339537, 7.47267751555080687676226229744, 8.435593919882535579859688276586