L(s) = 1 | − 14·2-s − 48·3-s + 68·4-s + 125·5-s + 672·6-s − 1.64e3·7-s + 840·8-s + 117·9-s − 1.75e3·10-s + 172·11-s − 3.26e3·12-s + 3.86e3·13-s + 2.30e4·14-s − 6.00e3·15-s − 2.04e4·16-s − 1.22e4·17-s − 1.63e3·18-s − 2.59e4·19-s + 8.50e3·20-s + 7.89e4·21-s − 2.40e3·22-s + 1.29e4·23-s − 4.03e4·24-s + 1.56e4·25-s − 5.40e4·26-s + 9.93e4·27-s − 1.11e5·28-s + ⋯ |
L(s) = 1 | − 1.23·2-s − 1.02·3-s + 0.531·4-s + 0.447·5-s + 1.27·6-s − 1.81·7-s + 0.580·8-s + 0.0534·9-s − 0.553·10-s + 0.0389·11-s − 0.545·12-s + 0.487·13-s + 2.24·14-s − 0.459·15-s − 1.24·16-s − 0.604·17-s − 0.0662·18-s − 0.867·19-s + 0.237·20-s + 1.85·21-s − 0.0482·22-s + 0.222·23-s − 0.595·24-s + 1/5·25-s − 0.603·26-s + 0.971·27-s − 0.962·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - p^{3} T \) |
good | 2 | \( 1 + 7 p T + p^{7} T^{2} \) |
| 3 | \( 1 + 16 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1644 T + p^{7} T^{2} \) |
| 11 | \( 1 - 172 T + p^{7} T^{2} \) |
| 13 | \( 1 - 3862 T + p^{7} T^{2} \) |
| 17 | \( 1 + 12254 T + p^{7} T^{2} \) |
| 19 | \( 1 + 25940 T + p^{7} T^{2} \) |
| 23 | \( 1 - 564 p T + p^{7} T^{2} \) |
| 29 | \( 1 + 81610 T + p^{7} T^{2} \) |
| 31 | \( 1 + 156888 T + p^{7} T^{2} \) |
| 37 | \( 1 - 110126 T + p^{7} T^{2} \) |
| 41 | \( 1 - 467882 T + p^{7} T^{2} \) |
| 43 | \( 1 + 499208 T + p^{7} T^{2} \) |
| 47 | \( 1 + 396884 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1280498 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1337420 T + p^{7} T^{2} \) |
| 61 | \( 1 + 923978 T + p^{7} T^{2} \) |
| 67 | \( 1 + 797304 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5103392 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4267478 T + p^{7} T^{2} \) |
| 79 | \( 1 + 960 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6140832 T + p^{7} T^{2} \) |
| 89 | \( 1 - 2010570 T + p^{7} T^{2} \) |
| 97 | \( 1 + 4881934 T + p^{7} T^{2} \) |
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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
−22.16548253268393328947846819386, −19.79127854535097455062440363103, −18.48308219402311061229510076046, −17.12302975315060672117957157880, −16.19722697511786235709360116310, −13.03645802025512111543084479615, −10.75638735439445487761160215799, −9.274009186839343165913297371932, −6.41815377421923413546724317339, 0,
6.41815377421923413546724317339, 9.274009186839343165913297371932, 10.75638735439445487761160215799, 13.03645802025512111543084479615, 16.19722697511786235709360116310, 17.12302975315060672117957157880, 18.48308219402311061229510076046, 19.79127854535097455062440363103, 22.16548253268393328947846819386