L(s) = 1 | − 2-s − 1.32·3-s + 4-s + 2.36·5-s + 1.32·6-s + 2.18·7-s − 8-s − 1.24·9-s − 2.36·10-s − 5.72·11-s − 1.32·12-s + 0.593·13-s − 2.18·14-s − 3.13·15-s + 16-s − 7.42·17-s + 1.24·18-s − 3.56·19-s + 2.36·20-s − 2.88·21-s + 5.72·22-s + 4.42·23-s + 1.32·24-s + 0.596·25-s − 0.593·26-s + 5.62·27-s + 2.18·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.763·3-s + 0.5·4-s + 1.05·5-s + 0.540·6-s + 0.824·7-s − 0.353·8-s − 0.416·9-s − 0.748·10-s − 1.72·11-s − 0.381·12-s + 0.164·13-s − 0.583·14-s − 0.808·15-s + 0.250·16-s − 1.80·17-s + 0.294·18-s − 0.817·19-s + 0.528·20-s − 0.630·21-s + 1.22·22-s + 0.922·23-s + 0.270·24-s + 0.119·25-s − 0.116·26-s + 1.08·27-s + 0.412·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 + 1.32T + 3T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 + 5.72T + 11T^{2} \) |
| 13 | \( 1 - 0.593T + 13T^{2} \) |
| 17 | \( 1 + 7.42T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 4.42T + 23T^{2} \) |
| 29 | \( 1 + 3.86T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 7.14T + 37T^{2} \) |
| 41 | \( 1 + 12.6T + 41T^{2} \) |
| 43 | \( 1 - 1.56T + 43T^{2} \) |
| 47 | \( 1 + 4.71T + 47T^{2} \) |
| 53 | \( 1 - 2.42T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 3.35T + 61T^{2} \) |
| 67 | \( 1 - 8.04T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 0.965T + 97T^{2} \) |
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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
−9.991198570023228661182429232030, −8.667009705040072003209182771105, −8.407115717887632952624664244946, −7.08444357024991670251947482116, −6.28153891949457198481846618303, −5.38280750115608869363478336265, −4.76405810025956420729923963557, −2.72382706081070357360295950535, −1.84495215660545507175918725981, 0,
1.84495215660545507175918725981, 2.72382706081070357360295950535, 4.76405810025956420729923963557, 5.38280750115608869363478336265, 6.28153891949457198481846618303, 7.08444357024991670251947482116, 8.407115717887632952624664244946, 8.667009705040072003209182771105, 9.991198570023228661182429232030