| L(s) = 1 | − 4.57·3-s + 18.9·7-s − 6.03·9-s + 7.94·11-s − 43.9·13-s − 124.·17-s − 93.4·19-s − 86.5·21-s − 23.6·23-s + 151.·27-s − 171.·29-s + 70.6·31-s − 36.3·33-s + 275.·37-s + 201.·39-s + 259.·41-s + 433.·43-s − 249.·47-s + 14.3·49-s + 571.·51-s − 112.·53-s + 428.·57-s − 844.·59-s + 768.·61-s − 114.·63-s − 385.·67-s + 108.·69-s + ⋯ |
| L(s) = 1 | − 0.881·3-s + 1.02·7-s − 0.223·9-s + 0.217·11-s − 0.937·13-s − 1.78·17-s − 1.12·19-s − 0.899·21-s − 0.214·23-s + 1.07·27-s − 1.09·29-s + 0.409·31-s − 0.191·33-s + 1.22·37-s + 0.825·39-s + 0.988·41-s + 1.53·43-s − 0.774·47-s + 0.0419·49-s + 1.56·51-s − 0.292·53-s + 0.994·57-s − 1.86·59-s + 1.61·61-s − 0.228·63-s − 0.703·67-s + 0.188·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9002044710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9002044710\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.57T + 27T^{2} \) |
| 7 | \( 1 - 18.9T + 343T^{2} \) |
| 11 | \( 1 - 7.94T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 23.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 70.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 275.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 433.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 112.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 844.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 768.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 385.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 350.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 773.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 497.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 349.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
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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
−8.791094661766210864236940982179, −8.018099150988790264979601568551, −7.17437278799578294374106224446, −6.30618288241717050710925948240, −5.68035404295747988064209562133, −4.57755874484070408720458883548, −4.37677439614562375534411755561, −2.66121439000545960618742331135, −1.83719217152007706646011939650, −0.44297662259879002997533149345,
0.44297662259879002997533149345, 1.83719217152007706646011939650, 2.66121439000545960618742331135, 4.37677439614562375534411755561, 4.57755874484070408720458883548, 5.68035404295747988064209562133, 6.30618288241717050710925948240, 7.17437278799578294374106224446, 8.018099150988790264979601568551, 8.791094661766210864236940982179
