L(s) = 1 | − 17.9·5-s − 7.28·7-s − 4.61·11-s + 29.8·13-s + 67.9·17-s − 111.·19-s + 218.·23-s + 198.·25-s + 34.2·29-s + 77.7·31-s + 130.·35-s − 347.·37-s + 234.·41-s − 53.3·43-s + 385.·47-s − 289.·49-s − 461.·53-s + 82.9·55-s − 7.16·59-s + 416.·61-s − 537.·65-s + 869.·67-s + 585.·71-s − 733.·73-s + 33.6·77-s − 1.17e3·79-s + 67.4·83-s + ⋯ |
L(s) = 1 | − 1.60·5-s − 0.393·7-s − 0.126·11-s + 0.637·13-s + 0.969·17-s − 1.34·19-s + 1.98·23-s + 1.58·25-s + 0.219·29-s + 0.450·31-s + 0.632·35-s − 1.54·37-s + 0.893·41-s − 0.189·43-s + 1.19·47-s − 0.845·49-s − 1.19·53-s + 0.203·55-s − 0.0158·59-s + 0.873·61-s − 1.02·65-s + 1.58·67-s + 0.979·71-s − 1.17·73-s + 0.0497·77-s − 1.66·79-s + 0.0891·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 17.9T + 125T^{2} \) |
| 7 | \( 1 + 7.28T + 343T^{2} \) |
| 11 | \( 1 + 4.61T + 1.33e3T^{2} \) |
| 13 | \( 1 - 29.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 218.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 77.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 347.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 53.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 385.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 461.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 7.16T + 2.05e5T^{2} \) |
| 61 | \( 1 - 416.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 869.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 585.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 733.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 67.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 965.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43T + 9.12e5T^{2} \) |
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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
−8.607338401087884938342869916459, −8.206456562935355403863327646561, −7.21270365519848023103082175173, −6.63596215680331023375663932097, −5.42269330141605612817562587745, −4.42231172351498814536599797808, −3.63204532354754052126710465899, −2.85690247342155759248614698861, −1.11204714564057794898328737805, 0,
1.11204714564057794898328737805, 2.85690247342155759248614698861, 3.63204532354754052126710465899, 4.42231172351498814536599797808, 5.42269330141605612817562587745, 6.63596215680331023375663932097, 7.21270365519848023103082175173, 8.206456562935355403863327646561, 8.607338401087884938342869916459