| L(s) = 1 | + 24.8·3-s − 84.4·5-s + 375.·9-s + 434.·11-s + 848.·13-s − 2.10e3·15-s − 301.·17-s − 2.90e3·19-s + 1.70e3·23-s + 4.01e3·25-s + 3.28e3·27-s + 3.84e3·29-s + 969.·31-s + 1.07e4·33-s + 2.50e3·37-s + 2.11e4·39-s − 1.32e4·41-s − 7.78e3·43-s − 3.16e4·45-s + 5.48e3·47-s − 7.49e3·51-s + 2.28e4·53-s − 3.66e4·55-s − 7.23e4·57-s + 1.11e4·59-s + 2.22e4·61-s − 7.17e4·65-s + ⋯ |
| L(s) = 1 | + 1.59·3-s − 1.51·5-s + 1.54·9-s + 1.08·11-s + 1.39·13-s − 2.41·15-s − 0.252·17-s − 1.84·19-s + 0.673·23-s + 1.28·25-s + 0.867·27-s + 0.848·29-s + 0.181·31-s + 1.72·33-s + 0.301·37-s + 2.22·39-s − 1.23·41-s − 0.641·43-s − 2.33·45-s + 0.362·47-s − 0.403·51-s + 1.11·53-s − 1.63·55-s − 2.94·57-s + 0.415·59-s + 0.764·61-s − 2.10·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.451497462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.451497462\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 24.8T + 243T^{2} \) |
| 5 | \( 1 + 84.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 434.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 848.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 301.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.90e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.70e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.84e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 969.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.32e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.78e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.48e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.21e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.91e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 64.5T + 8.58e9T^{2} \) |
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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
−9.101781809232765092865448746299, −8.480054691105504153956893687560, −8.259413793500820924326782251824, −7.09929938291988336435846035117, −6.41554858615778695448166007956, −4.51256020725086017001100173135, −3.84622129825668948234737250928, −3.31098903197883027925775373629, −2.03591444958387339069535289665, −0.803771445583756165431061014347,
0.803771445583756165431061014347, 2.03591444958387339069535289665, 3.31098903197883027925775373629, 3.84622129825668948234737250928, 4.51256020725086017001100173135, 6.41554858615778695448166007956, 7.09929938291988336435846035117, 8.259413793500820924326782251824, 8.480054691105504153956893687560, 9.101781809232765092865448746299
