| L(s) = 1 | + 2.35·3-s + 5-s − 2.89·7-s + 2.53·9-s − 2.60·11-s + 4.35·13-s + 2.35·15-s − 7.29·17-s − 1.74·19-s − 6.80·21-s − 4.60·23-s + 25-s − 1.09·27-s + 6.00·29-s − 2.06·31-s − 6.13·33-s − 2.89·35-s + 1.66·37-s + 10.2·39-s − 4.61·41-s − 4.28·43-s + 2.53·45-s − 11.7·47-s + 1.37·49-s − 17.1·51-s + 3.86·53-s − 2.60·55-s + ⋯ |
| L(s) = 1 | + 1.35·3-s + 0.447·5-s − 1.09·7-s + 0.845·9-s − 0.786·11-s + 1.20·13-s + 0.607·15-s − 1.77·17-s − 0.400·19-s − 1.48·21-s − 0.960·23-s + 0.200·25-s − 0.210·27-s + 1.11·29-s − 0.370·31-s − 1.06·33-s − 0.489·35-s + 0.274·37-s + 1.64·39-s − 0.720·41-s − 0.653·43-s + 0.377·45-s − 1.70·47-s + 0.196·49-s − 2.40·51-s + 0.530·53-s − 0.351·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 - 4.35T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 - 6.00T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 3.86T + 53T^{2} \) |
| 59 | \( 1 - 4.40T + 59T^{2} \) |
| 61 | \( 1 - 3.32T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 3.25T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.113T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.284403885757295485241554676185, −7.11339156646636851406767838684, −6.46705779393457840471707269631, −5.92197915841313797000415796664, −4.74972860648356196304480934325, −3.89183626377364171021660879363, −3.18200950024979271352563049185, −2.49767536169482682059987469536, −1.73190225661377602836631417944, 0,
1.73190225661377602836631417944, 2.49767536169482682059987469536, 3.18200950024979271352563049185, 3.89183626377364171021660879363, 4.74972860648356196304480934325, 5.92197915841313797000415796664, 6.46705779393457840471707269631, 7.11339156646636851406767838684, 8.284403885757295485241554676185
