| L(s) = 1 | + 3.24·2-s − 5.49·3-s + 2.53·4-s − 16.0·5-s − 17.8·6-s + 7·7-s − 17.7·8-s + 3.16·9-s − 52.2·10-s − 13.9·12-s − 35.3·13-s + 22.7·14-s + 88.4·15-s − 77.8·16-s − 40.4·17-s + 10.2·18-s − 118.·19-s − 40.7·20-s − 38.4·21-s − 174.·23-s + 97.4·24-s + 134.·25-s − 114.·26-s + 130.·27-s + 17.7·28-s + 262.·29-s + 286.·30-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.05·3-s + 0.316·4-s − 1.43·5-s − 1.21·6-s + 0.377·7-s − 0.784·8-s + 0.117·9-s − 1.65·10-s − 0.334·12-s − 0.754·13-s + 0.433·14-s + 1.52·15-s − 1.21·16-s − 0.577·17-s + 0.134·18-s − 1.42·19-s − 0.455·20-s − 0.399·21-s − 1.58·23-s + 0.828·24-s + 1.07·25-s − 0.865·26-s + 0.933·27-s + 0.119·28-s + 1.68·29-s + 1.74·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5166195726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5166195726\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3.24T + 8T^{2} \) |
| 3 | \( 1 + 5.49T + 27T^{2} \) |
| 5 | \( 1 + 16.0T + 125T^{2} \) |
| 13 | \( 1 + 35.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 40.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 174.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 262.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 36.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 287.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 272.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 507.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 35.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 979.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 750.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 395.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 582.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 806.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 957.T + 9.12e5T^{2} \) |
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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
−10.15364617340704496479691036112, −8.692165156886550043103518769685, −8.086766936372657198370434050822, −6.85499734329122203101798381548, −6.19087026861756053298857249678, −5.11961573762048942432391398474, −4.47912252751872200064768012615, −3.83635299478462737059996578924, −2.48715025033123340933215350948, −0.33210805545570359202129519817,
0.33210805545570359202129519817, 2.48715025033123340933215350948, 3.83635299478462737059996578924, 4.47912252751872200064768012615, 5.11961573762048942432391398474, 6.19087026861756053298857249678, 6.85499734329122203101798381548, 8.086766936372657198370434050822, 8.692165156886550043103518769685, 10.15364617340704496479691036112
