| L(s) = 1 | + 4.70·2-s + 14.1·4-s + 16.2·7-s + 28.7·8-s + 40.2·11-s − 19.7·13-s + 76.2·14-s + 22.1·16-s + 83.0·17-s − 48.8·19-s + 189.·22-s + 1.61·23-s − 93.0·26-s + 228.·28-s + 24.5·29-s − 12.4·31-s − 125.·32-s + 390.·34-s − 325.·37-s − 229.·38-s + 242.·41-s − 367.·43-s + 567.·44-s + 7.58·46-s − 204.·47-s − 80.2·49-s − 279.·52-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.76·4-s + 0.875·7-s + 1.26·8-s + 1.10·11-s − 0.422·13-s + 1.45·14-s + 0.345·16-s + 1.18·17-s − 0.589·19-s + 1.83·22-s + 0.0146·23-s − 0.701·26-s + 1.54·28-s + 0.157·29-s − 0.0719·31-s − 0.694·32-s + 1.96·34-s − 1.44·37-s − 0.980·38-s + 0.923·41-s − 1.30·43-s + 1.94·44-s + 0.0242·46-s − 0.634·47-s − 0.233·49-s − 0.744·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.945550844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.945550844\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.70T + 8T^{2} \) |
| 7 | \( 1 - 16.2T + 343T^{2} \) |
| 11 | \( 1 - 40.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.61T + 1.21e4T^{2} \) |
| 29 | \( 1 - 24.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 325.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 204.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 61.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 477.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 558.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 558.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 96.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−11.94346387860994636978927591178, −11.36947303961115371933134026381, −10.15541839878537910026852799382, −8.729182469611316358590114799750, −7.42990441439697567149532125474, −6.37959091967386491274566000400, −5.30829016090893169540249818246, −4.39580327020140988465203035346, −3.30170718230716042603967448156, −1.72923619680681977535598499642,
1.72923619680681977535598499642, 3.30170718230716042603967448156, 4.39580327020140988465203035346, 5.30829016090893169540249818246, 6.37959091967386491274566000400, 7.42990441439697567149532125474, 8.729182469611316358590114799750, 10.15541839878537910026852799382, 11.36947303961115371933134026381, 11.94346387860994636978927591178
