| L(s) = 1 | − 2.21·2-s − 3.10·4-s + 17.1·7-s + 24.5·8-s + 66.8·11-s + 72.7·13-s − 37.8·14-s − 29.5·16-s + 40.2·17-s + 38.4·19-s − 147.·22-s − 204.·23-s − 160.·26-s − 53.1·28-s − 21.6·29-s + 128.·31-s − 131.·32-s − 89.1·34-s + 19.4·37-s − 85.1·38-s + 270.·41-s + 242.·43-s − 207.·44-s + 452.·46-s − 307.·47-s − 50.4·49-s − 225.·52-s + ⋯ |
| L(s) = 1 | − 0.782·2-s − 0.388·4-s + 0.923·7-s + 1.08·8-s + 1.83·11-s + 1.55·13-s − 0.722·14-s − 0.461·16-s + 0.574·17-s + 0.464·19-s − 1.43·22-s − 1.85·23-s − 1.21·26-s − 0.358·28-s − 0.138·29-s + 0.743·31-s − 0.724·32-s − 0.449·34-s + 0.0862·37-s − 0.363·38-s + 1.02·41-s + 0.861·43-s − 0.711·44-s + 1.45·46-s − 0.954·47-s − 0.147·49-s − 0.602·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.664303431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664303431\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.21T + 8T^{2} \) |
| 7 | \( 1 - 17.1T + 343T^{2} \) |
| 11 | \( 1 - 66.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 204.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 21.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 270.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 242.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 289.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 17.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 764.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 532.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 220.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 253.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 457.T + 9.12e5T^{2} \) |
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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
−9.857082969120103232710642729312, −9.242794508073275413835668920266, −8.332140129212417423830752980090, −7.909710992568783224963466367072, −6.62341989012481237227049336553, −5.69071448649304152314363384776, −4.35085859110339764586910181187, −3.74113717972255161309557391414, −1.66186192662318982679702213669, −0.970705560094785034991030399450,
0.970705560094785034991030399450, 1.66186192662318982679702213669, 3.74113717972255161309557391414, 4.35085859110339764586910181187, 5.69071448649304152314363384776, 6.62341989012481237227049336553, 7.909710992568783224963466367072, 8.332140129212417423830752980090, 9.242794508073275413835668920266, 9.857082969120103232710642729312
