| L(s) = 1 | + 3-s + 3·5-s − 2·9-s − 3·11-s − 13-s + 3·15-s − 3·17-s + 5·19-s + 9·23-s + 4·25-s − 5·27-s − 6·29-s − 31-s − 3·33-s − 7·37-s − 39-s − 6·41-s − 8·43-s − 6·45-s − 3·47-s − 3·51-s − 9·53-s − 9·55-s + 5·57-s − 9·59-s − 11·61-s − 3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 2/3·9-s − 0.904·11-s − 0.277·13-s + 0.774·15-s − 0.727·17-s + 1.14·19-s + 1.87·23-s + 4/5·25-s − 0.962·27-s − 1.11·29-s − 0.179·31-s − 0.522·33-s − 1.15·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s − 0.894·45-s − 0.437·47-s − 0.420·51-s − 1.23·53-s − 1.21·55-s + 0.662·57-s − 1.17·59-s − 1.40·61-s − 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−17.00330142006234, −16.50866134255382, −15.43729336272166, −15.30016363541951, −14.50148955992239, −13.87572769834727, −13.60472278543070, −13.03107918587121, −12.52608106126585, −11.50514642081500, −11.06374364679551, −10.39954995848631, −9.667248517469841, −9.288026610066365, −8.732757075206851, −8.023854287119902, −7.286819147038066, −6.657467581051407, −5.847160634872331, −5.172384376952671, −4.911735856047732, −3.400507565588069, −3.009675756663938, −2.180676537563386, −1.512937742005358, 0,
1.512937742005358, 2.180676537563386, 3.009675756663938, 3.400507565588069, 4.911735856047732, 5.172384376952671, 5.847160634872331, 6.657467581051407, 7.286819147038066, 8.023854287119902, 8.732757075206851, 9.288026610066365, 9.667248517469841, 10.39954995848631, 11.06374364679551, 11.50514642081500, 12.52608106126585, 13.03107918587121, 13.60472278543070, 13.87572769834727, 14.50148955992239, 15.30016363541951, 15.43729336272166, 16.50866134255382, 17.00330142006234
