| L(s) = 1 | + 4.23·2-s + 9.91·4-s − 2.28·7-s + 8.11·8-s − 60.7·11-s + 64.3·13-s − 9.68·14-s − 44.9·16-s + 70.6·17-s − 150.·19-s − 257.·22-s + 59.6·23-s + 272.·26-s − 22.6·28-s − 56.1·29-s − 156.·31-s − 255.·32-s + 298.·34-s − 192.·37-s − 639.·38-s − 119.·41-s − 187.·43-s − 602.·44-s + 252.·46-s + 12.8·47-s − 337.·49-s + 637.·52-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.23·4-s − 0.123·7-s + 0.358·8-s − 1.66·11-s + 1.37·13-s − 0.184·14-s − 0.703·16-s + 1.00·17-s − 1.82·19-s − 2.49·22-s + 0.540·23-s + 2.05·26-s − 0.153·28-s − 0.359·29-s − 0.907·31-s − 1.41·32-s + 1.50·34-s − 0.855·37-s − 2.72·38-s − 0.456·41-s − 0.665·43-s − 2.06·44-s + 0.808·46-s + 0.0398·47-s − 0.984·49-s + 1.70·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1125 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 4.23T + 8T^{2} \) |
| 7 | \( 1 + 2.28T + 343T^{2} \) |
| 11 | \( 1 + 60.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 59.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 56.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 192.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 119.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 187.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 12.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 61.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 257.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 555.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 891.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 489.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 761.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 126.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−8.870528733002511064375982491607, −8.161889034652121000364930262323, −7.10043171556648971291257617074, −6.16393665717923325526458088424, −5.52497716369609014403195955304, −4.74888000356753170856192772137, −3.71202489330034462909421683833, −3.01013778040765422522186783016, −1.86558566252053172453372502856, 0,
1.86558566252053172453372502856, 3.01013778040765422522186783016, 3.71202489330034462909421683833, 4.74888000356753170856192772137, 5.52497716369609014403195955304, 6.16393665717923325526458088424, 7.10043171556648971291257617074, 8.161889034652121000364930262323, 8.870528733002511064375982491607
