| L(s) = 1 | + 3·3-s − 19.8·5-s − 19.8·7-s + 9·9-s − 48·11-s − 79.5·13-s − 59.6·15-s − 42·17-s + 92·19-s − 59.6·21-s + 39.7·23-s + 271·25-s + 27·27-s + 19.8·29-s + 139.·31-s − 144·33-s + 396·35-s − 198.·37-s − 238.·39-s + 6·41-s + 92·43-s − 179.·45-s − 39.7·47-s + 53·49-s − 126·51-s + 497.·53-s + 955.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.77·5-s − 1.07·7-s + 0.333·9-s − 1.31·11-s − 1.69·13-s − 1.02·15-s − 0.599·17-s + 1.11·19-s − 0.620·21-s + 0.360·23-s + 2.16·25-s + 0.192·27-s + 0.127·29-s + 0.807·31-s − 0.759·33-s + 1.91·35-s − 0.884·37-s − 0.980·39-s + 0.0228·41-s + 0.326·43-s − 0.593·45-s − 0.123·47-s + 0.154·49-s − 0.345·51-s + 1.28·53-s + 2.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6077147236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6077147236\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 + 19.8T + 125T^{2} \) |
| 7 | \( 1 + 19.8T + 343T^{2} \) |
| 11 | \( 1 + 48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42T + 4.91e3T^{2} \) |
| 19 | \( 1 - 92T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 19.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92T + 7.95e4T^{2} \) |
| 47 | \( 1 + 39.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 497.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 516T + 2.05e5T^{2} \) |
| 61 | \( 1 - 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 524T + 3.00e5T^{2} \) |
| 71 | \( 1 + 994.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 430T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 432T + 5.71e5T^{2} \) |
| 89 | \( 1 + 630T + 7.04e5T^{2} \) |
| 97 | \( 1 - 862T + 9.12e5T^{2} \) |
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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
−9.923197924784733136958132195661, −9.020618563979615182111949787431, −8.034854300454489148450223921630, −7.44145408819017808587841127649, −6.85812472647848799574257023941, −5.20796605339152220729150880329, −4.36757084190291984414072351430, −3.23031506611215807303950327115, −2.66409682345466495325816226579, −0.39940529911352392591634383942,
0.39940529911352392591634383942, 2.66409682345466495325816226579, 3.23031506611215807303950327115, 4.36757084190291984414072351430, 5.20796605339152220729150880329, 6.85812472647848799574257023941, 7.44145408819017808587841127649, 8.034854300454489148450223921630, 9.020618563979615182111949787431, 9.923197924784733136958132195661
