| L(s) = 1 | − 19.3·3-s + 49.9·5-s + 129.·9-s + 351.·11-s + 853.·13-s − 965.·15-s + 2.26e3·17-s − 201.·19-s + 1.06e3·23-s − 626.·25-s + 2.18e3·27-s + 6.86e3·29-s − 5.40e3·31-s − 6.78e3·33-s + 9.91e3·37-s − 1.64e4·39-s − 9.38e3·41-s + 2.23e4·43-s + 6.49e3·45-s + 6.79e3·47-s − 4.37e4·51-s + 3.49e3·53-s + 1.75e4·55-s + 3.89e3·57-s − 2.75e4·59-s − 2.89e4·61-s + 4.26e4·65-s + ⋯ |
| L(s) = 1 | − 1.23·3-s + 0.894·5-s + 0.534·9-s + 0.875·11-s + 1.40·13-s − 1.10·15-s + 1.90·17-s − 0.128·19-s + 0.421·23-s − 0.200·25-s + 0.576·27-s + 1.51·29-s − 1.00·31-s − 1.08·33-s + 1.19·37-s − 1.73·39-s − 0.871·41-s + 1.84·43-s + 0.477·45-s + 0.448·47-s − 2.35·51-s + 0.170·53-s + 0.782·55-s + 0.158·57-s − 1.03·59-s − 0.994·61-s + 1.25·65-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
| \(L(3)\) |
\(\approx\) |
\(2.269117349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269117349\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 19.3T + 243T^{2} \) |
| 5 | \( 1 - 49.9T + 3.12e3T^{2} \) |
| 11 | \( 1 - 351.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 853.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.26e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 201.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.91e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.38e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.23e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.79e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.49e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.35e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.18e3T + 8.58e9T^{2} \) |
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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
−9.649881901846284324395826257102, −8.860528617990001692824111285454, −7.73508463288395076529374466314, −6.52897561037627444468554703337, −5.97406284070736939913274617091, −5.44216357872796234196502551338, −4.24254585814476366165236946168, −3.07624201199237541957761594002, −1.45307887881390729921887484298, −0.843339614166417512697578810747,
0.843339614166417512697578810747, 1.45307887881390729921887484298, 3.07624201199237541957761594002, 4.24254585814476366165236946168, 5.44216357872796234196502551338, 5.97406284070736939913274617091, 6.52897561037627444468554703337, 7.73508463288395076529374466314, 8.860528617990001692824111285454, 9.649881901846284324395826257102
