| L(s) = 1 | − 25·5-s + 101.·7-s + 254.·11-s + 1.12e3·13-s + 2.16e3·17-s + 486.·19-s − 231.·23-s + 625·25-s + 8.56e3·29-s − 763.·31-s − 2.53e3·35-s + 3.23e3·37-s + 4.42e3·41-s − 1.16e4·43-s + 1.40e4·47-s − 6.50e3·49-s − 2.53e4·53-s − 6.35e3·55-s + 1.48e4·59-s − 1.81e4·61-s − 2.80e4·65-s − 8.82e3·67-s − 2.48e4·71-s + 3.82e4·73-s + 2.57e4·77-s − 7.19e4·79-s + 5.72e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.782·7-s + 0.633·11-s + 1.83·13-s + 1.81·17-s + 0.308·19-s − 0.0912·23-s + 0.200·25-s + 1.89·29-s − 0.142·31-s − 0.350·35-s + 0.388·37-s + 0.411·41-s − 0.963·43-s + 0.925·47-s − 0.387·49-s − 1.23·53-s − 0.283·55-s + 0.553·59-s − 0.622·61-s − 0.822·65-s − 0.240·67-s − 0.586·71-s + 0.839·73-s + 0.495·77-s − 1.29·79-s + 0.912·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.362441450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.362441450\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| good | 7 | \( 1 - 101.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 254.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 486.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 231.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 763.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.42e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.40e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.53e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.48e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.81e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.82e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.29e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.20e4T + 8.58e9T^{2} \) |
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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
−8.987990876468741029543125291244, −8.246027588514404011925770051671, −7.72098364954885882731668851617, −6.56756407028946367000933148081, −5.80503540716647383357228377663, −4.78833342257178251974160703740, −3.82781313017088941882521346465, −3.07490381312605340948849636452, −1.45018873625221503650722618943, −0.920887103915179349423752461320,
0.920887103915179349423752461320, 1.45018873625221503650722618943, 3.07490381312605340948849636452, 3.82781313017088941882521346465, 4.78833342257178251974160703740, 5.80503540716647383357228377663, 6.56756407028946367000933148081, 7.72098364954885882731668851617, 8.246027588514404011925770051671, 8.987990876468741029543125291244
