| L(s) = 1 | − 338.·5-s − 1.29e3·7-s + 3.71e3·11-s − 1.34e4·13-s − 2.65e3·17-s − 2.97e4·19-s − 5.18e4·23-s + 3.66e4·25-s − 1.74e5·29-s − 3.16e5·31-s + 4.37e5·35-s − 3.30e5·37-s − 3.32e5·41-s + 3.45e5·43-s + 1.03e6·47-s + 8.43e5·49-s + 1.09e6·53-s − 1.25e6·55-s + 8.45e5·59-s + 4.15e5·61-s + 4.56e6·65-s − 3.09e6·67-s + 1.28e6·71-s + 3.30e6·73-s − 4.79e6·77-s − 4.15e6·79-s − 1.00e7·83-s + ⋯ |
| L(s) = 1 | − 1.21·5-s − 1.42·7-s + 0.841·11-s − 1.70·13-s − 0.131·17-s − 0.993·19-s − 0.888·23-s + 0.468·25-s − 1.33·29-s − 1.90·31-s + 1.72·35-s − 1.07·37-s − 0.753·41-s + 0.662·43-s + 1.45·47-s + 1.02·49-s + 1.00·53-s − 1.02·55-s + 0.535·59-s + 0.234·61-s + 2.06·65-s − 1.25·67-s + 0.425·71-s + 0.993·73-s − 1.19·77-s − 0.949·79-s − 1.93·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1283374994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1283374994\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 338.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.29e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.71e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.34e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.65e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.97e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.18e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.74e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.16e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.30e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.32e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.03e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.09e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.45e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.15e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.09e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.30e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.00e7T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.59e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.07e6T + 8.07e13T^{2} \) |
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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
−10.54390957381731963631404615758, −9.587584953945587640155459417189, −8.795191336279749806185199746709, −7.44466900477756414320529318013, −6.93544428205912886202197951444, −5.67345178085652244647902790776, −4.17648501916612511134460696409, −3.54296245177450135323876222097, −2.17162176303531877778532796257, −0.16448116578796171686886981685,
0.16448116578796171686886981685, 2.17162176303531877778532796257, 3.54296245177450135323876222097, 4.17648501916612511134460696409, 5.67345178085652244647902790776, 6.93544428205912886202197951444, 7.44466900477756414320529318013, 8.795191336279749806185199746709, 9.587584953945587640155459417189, 10.54390957381731963631404615758
