| L(s) = 1 | + 16.3·3-s + 51.0·5-s + 24.8·9-s − 273.·11-s − 369.·13-s + 834.·15-s − 1.03e3·17-s − 2.18e3·19-s − 1.01e3·23-s − 522.·25-s − 3.57e3·27-s − 212.·29-s + 229.·31-s − 4.47e3·33-s − 9.11e3·37-s − 6.04e3·39-s − 1.67e4·41-s + 1.61e4·43-s + 1.26e3·45-s + 1.12e3·47-s − 1.69e4·51-s + 3.41e4·53-s − 1.39e4·55-s − 3.57e4·57-s + 1.11e4·59-s + 3.59e4·61-s − 1.88e4·65-s + ⋯ |
| L(s) = 1 | + 1.04·3-s + 0.912·5-s + 0.102·9-s − 0.681·11-s − 0.605·13-s + 0.958·15-s − 0.869·17-s − 1.38·19-s − 0.399·23-s − 0.167·25-s − 0.942·27-s − 0.0469·29-s + 0.0429·31-s − 0.715·33-s − 1.09·37-s − 0.636·39-s − 1.55·41-s + 1.32·43-s + 0.0933·45-s + 0.0743·47-s − 0.913·51-s + 1.67·53-s − 0.621·55-s − 1.45·57-s + 0.418·59-s + 1.23·61-s − 0.552·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 16.3T + 243T^{2} \) |
| 5 | \( 1 - 51.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 273.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 369.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.01e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 212.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 229.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.67e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.64e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.52e4T + 8.58e9T^{2} \) |
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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
−9.980373320560426176400257852348, −8.982870480470943009933018677784, −8.409459170064418810884860828897, −7.32439579913366078392983698209, −6.21802531278008805373635084194, −5.15937049155712753270325705560, −3.88187887281872825126218783342, −2.50647744687863185405074084861, −2.01168022879392590688964391286, 0,
2.01168022879392590688964391286, 2.50647744687863185405074084861, 3.88187887281872825126218783342, 5.15937049155712753270325705560, 6.21802531278008805373635084194, 7.32439579913366078392983698209, 8.409459170064418810884860828897, 8.982870480470943009933018677784, 9.980373320560426176400257852348
