| L(s) = 1 | − 202.·5-s − 343·7-s + 7.52e3·11-s − 1.70e3·13-s + 2.48e3·17-s − 3.98e4·19-s − 2.55e4·23-s − 3.71e4·25-s − 1.48e5·29-s − 3.35e4·31-s + 6.94e4·35-s + 4.00e5·37-s + 3.62e5·41-s − 3.24e5·43-s + 7.08e5·47-s + 1.17e5·49-s + 1.85e5·53-s − 1.52e6·55-s + 1.19e6·59-s + 2.51e6·61-s + 3.44e5·65-s − 2.85e6·67-s − 3.22e6·71-s + 5.01e6·73-s − 2.58e6·77-s + 5.94e6·79-s + 1.02e7·83-s + ⋯ |
| L(s) = 1 | − 0.724·5-s − 0.377·7-s + 1.70·11-s − 0.214·13-s + 0.122·17-s − 1.33·19-s − 0.438·23-s − 0.475·25-s − 1.13·29-s − 0.202·31-s + 0.273·35-s + 1.30·37-s + 0.821·41-s − 0.622·43-s + 0.995·47-s + 0.142·49-s + 0.171·53-s − 1.23·55-s + 0.756·59-s + 1.41·61-s + 0.155·65-s − 1.16·67-s − 1.06·71-s + 1.50·73-s − 0.644·77-s + 1.35·79-s + 1.95·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.553056300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553056300\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 343T \) |
| good | 5 | \( 1 + 202.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 7.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.70e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.48e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.98e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.55e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.35e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.24e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.08e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.85e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.19e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.51e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.01e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.94e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.02e7T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.85e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.52e7T + 8.07e13T^{2} \) |
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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
−10.95198485763641273762582184256, −9.712273205471157723709648294335, −8.924815097104136382331295828057, −7.84999593016287602422788543907, −6.80646036037028168159436303779, −5.91787430576143054860462718506, −4.27888304572557853056040212740, −3.67982727526506621342290639050, −2.07049225056689533275357687962, −0.62748393223175989382611561939,
0.62748393223175989382611561939, 2.07049225056689533275357687962, 3.67982727526506621342290639050, 4.27888304572557853056040212740, 5.91787430576143054860462718506, 6.80646036037028168159436303779, 7.84999593016287602422788543907, 8.924815097104136382331295828057, 9.712273205471157723709648294335, 10.95198485763641273762582184256
