| L(s) = 1 | + 21.2·3-s − 140.·7-s + 206.·9-s + 121·11-s + 1.07e3·13-s − 2.02e3·17-s + 79.8·19-s − 2.98e3·21-s + 96.0·23-s − 764.·27-s + 1.69e3·29-s + 2.33e3·31-s + 2.56e3·33-s + 6.71e3·37-s + 2.27e4·39-s + 1.99e4·41-s + 3.48e3·43-s + 7.98e3·47-s + 3.06e3·49-s − 4.28e4·51-s − 3.28e4·53-s + 1.69e3·57-s − 4.79e4·59-s + 1.99e4·61-s − 2.91e4·63-s + 2.24e4·67-s + 2.03e3·69-s + ⋯ |
| L(s) = 1 | + 1.36·3-s − 1.08·7-s + 0.851·9-s + 0.301·11-s + 1.75·13-s − 1.69·17-s + 0.0507·19-s − 1.47·21-s + 0.0378·23-s − 0.201·27-s + 0.373·29-s + 0.435·31-s + 0.410·33-s + 0.806·37-s + 2.39·39-s + 1.85·41-s + 0.287·43-s + 0.527·47-s + 0.182·49-s − 2.30·51-s − 1.60·53-s + 0.0690·57-s − 1.79·59-s + 0.685·61-s − 0.925·63-s + 0.611·67-s + 0.0515·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.524803534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.524803534\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 - 21.2T + 243T^{2} \) |
| 7 | \( 1 + 140.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.07e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.02e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 79.8T + 2.47e6T^{2} \) |
| 23 | \( 1 - 96.0T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.99e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.48e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.79e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 9.67e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.79e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.60e4T + 8.58e9T^{2} \) |
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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
−9.164643908543061920431643253024, −8.459381356217162728549569921863, −7.67397776338948057449731722160, −6.50034356767374029387290476495, −6.12946618679861536854256726970, −4.44516207644179983788323522076, −3.67818527367736416934057964646, −2.93866479704281369527279350870, −2.02184468734267250435254703291, −0.74209466607610779834925408269,
0.74209466607610779834925408269, 2.02184468734267250435254703291, 2.93866479704281369527279350870, 3.67818527367736416934057964646, 4.44516207644179983788323522076, 6.12946618679861536854256726970, 6.50034356767374029387290476495, 7.67397776338948057449731722160, 8.459381356217162728549569921863, 9.164643908543061920431643253024
