| L(s) = 1 | + 2.39·3-s + 3.28·5-s − 33.9·7-s − 21.2·9-s + 14.5·11-s + 86.5·13-s + 7.86·15-s + 102.·17-s − 105.·19-s − 81.3·21-s + 135.·23-s − 114.·25-s − 115.·27-s − 29·29-s − 223.·31-s + 34.9·33-s − 111.·35-s + 239.·37-s + 207.·39-s + 219.·41-s + 18.9·43-s − 69.7·45-s − 147.·47-s + 809.·49-s + 245.·51-s − 613.·53-s + 47.8·55-s + ⋯ |
| L(s) = 1 | + 0.461·3-s + 0.293·5-s − 1.83·7-s − 0.787·9-s + 0.399·11-s + 1.84·13-s + 0.135·15-s + 1.46·17-s − 1.27·19-s − 0.845·21-s + 1.22·23-s − 0.913·25-s − 0.824·27-s − 0.185·29-s − 1.29·31-s + 0.184·33-s − 0.537·35-s + 1.06·37-s + 0.851·39-s + 0.835·41-s + 0.0673·43-s − 0.230·45-s − 0.459·47-s + 2.35·49-s + 0.674·51-s − 1.58·53-s + 0.117·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 2.39T + 27T^{2} \) |
| 5 | \( 1 - 3.28T + 125T^{2} \) |
| 7 | \( 1 + 33.9T + 343T^{2} \) |
| 11 | \( 1 - 14.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 135.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 239.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 219.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 18.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 147.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 613.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 184.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 328.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 5.15T + 3.57e5T^{2} \) |
| 73 | \( 1 + 428.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 392.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 811.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 11.3T + 9.12e5T^{2} \) |
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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
−8.652701920207818468837729096055, −7.81544312908712416574292204145, −6.71107835708330829853372525011, −6.06563118126783771475488553755, −5.62929289204703623777309127067, −3.90939656418521474598996414243, −3.44471252786275419054327131019, −2.64409429148702377236338971480, −1.26502932532872807363026550251, 0,
1.26502932532872807363026550251, 2.64409429148702377236338971480, 3.44471252786275419054327131019, 3.90939656418521474598996414243, 5.62929289204703623777309127067, 6.06563118126783771475488553755, 6.71107835708330829853372525011, 7.81544312908712416574292204145, 8.652701920207818468837729096055
