| L(s) = 1 | + 4.91·2-s + 16.1·4-s + 5·5-s + 40.1·8-s + 24.5·10-s − 11.2·11-s − 31.5·13-s + 68.0·16-s + 67.9·17-s + 158.·19-s + 80.8·20-s − 55.0·22-s − 41.6·23-s + 25·25-s − 155.·26-s + 278.·29-s + 35.4·31-s + 13.3·32-s + 334.·34-s − 35.5·37-s + 778.·38-s + 200.·40-s − 154.·41-s − 309.·43-s − 181.·44-s − 204.·46-s + 277.·47-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.02·4-s + 0.447·5-s + 1.77·8-s + 0.777·10-s − 0.307·11-s − 0.673·13-s + 1.06·16-s + 0.970·17-s + 1.91·19-s + 0.903·20-s − 0.533·22-s − 0.378·23-s + 0.200·25-s − 1.17·26-s + 1.78·29-s + 0.205·31-s + 0.0735·32-s + 1.68·34-s − 0.157·37-s + 3.32·38-s + 0.793·40-s − 0.588·41-s − 1.09·43-s − 0.620·44-s − 0.657·46-s + 0.859·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.150193899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.150193899\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.91T + 8T^{2} \) |
| 11 | \( 1 + 11.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 158.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 35.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 35.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 309.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 79.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 901.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 514.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 774.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 697.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 441.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 305.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 925.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 46.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 779.T + 9.12e5T^{2} \) |
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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
−8.545007667596842795443620804789, −7.57241017584103997469838721731, −6.91795077661738455373510504233, −6.09773922532841440499698581917, −5.19108653255088530237515353122, −5.02688896572114882219457928835, −3.78140585033466480384801039899, −3.04451295539954882201981565478, −2.28986089814839037503177917420, −1.02797862745646075823150411519,
1.02797862745646075823150411519, 2.28986089814839037503177917420, 3.04451295539954882201981565478, 3.78140585033466480384801039899, 5.02688896572114882219457928835, 5.19108653255088530237515353122, 6.09773922532841440499698581917, 6.91795077661738455373510504233, 7.57241017584103997469838721731, 8.545007667596842795443620804789
