| L(s) = 1 | + 1.61·2-s + 0.618·4-s − 2.23·7-s − 2.23·8-s + 4·11-s + 2.47·13-s − 3.61·14-s − 4.85·16-s − 3.23·17-s − 19-s + 6.47·22-s − 1.23·23-s + 4.00·26-s − 1.38·28-s + 1.47·29-s − 1.52·31-s − 3.38·32-s − 5.23·34-s − 7.23·37-s − 1.61·38-s + 5·41-s − 4·43-s + 2.47·44-s − 2.00·46-s − 8.47·47-s − 1.99·49-s + 1.52·52-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s − 0.845·7-s − 0.790·8-s + 1.20·11-s + 0.685·13-s − 0.966·14-s − 1.21·16-s − 0.784·17-s − 0.229·19-s + 1.37·22-s − 0.257·23-s + 0.784·26-s − 0.261·28-s + 0.273·29-s − 0.274·31-s − 0.597·32-s − 0.897·34-s − 1.18·37-s − 0.262·38-s + 0.780·41-s − 0.609·43-s + 0.372·44-s − 0.294·46-s − 1.23·47-s − 0.285·49-s + 0.211·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 7.23T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 + 5T + 53T^{2} \) |
| 59 | \( 1 - 1.29T + 59T^{2} \) |
| 61 | \( 1 + 9.94T + 61T^{2} \) |
| 67 | \( 1 + 6.94T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 5.47T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 7.94T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.056023327232767146475966090313, −6.70175634470167584066178754424, −6.58452533513727896858657922304, −5.83108768853317082074619169742, −4.93472259812161407210469440139, −4.12010297425212673411718939321, −3.59885387423298458605657251673, −2.83461254356105916826486156371, −1.59558322377755067956703079596, 0,
1.59558322377755067956703079596, 2.83461254356105916826486156371, 3.59885387423298458605657251673, 4.12010297425212673411718939321, 4.93472259812161407210469440139, 5.83108768853317082074619169742, 6.58452533513727896858657922304, 6.70175634470167584066178754424, 8.056023327232767146475966090313
