| L(s) = 1 | + 25·5-s − 160.·7-s − 279.·11-s − 541.·13-s + 777.·17-s + 2.68e3·19-s + 3.69e3·23-s + 625·25-s + 8.35e3·29-s + 262.·31-s − 4.01e3·35-s − 1.49e4·37-s − 7.98e3·41-s − 5.13e3·43-s − 1.05e4·47-s + 8.92e3·49-s + 2.10e4·53-s − 6.98e3·55-s − 2.56e4·59-s + 8.19e3·61-s − 1.35e4·65-s − 5.19e4·67-s − 2.36e4·71-s + 2.09e4·73-s + 4.48e4·77-s + 3.92e4·79-s − 3.59e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.23·7-s − 0.696·11-s − 0.888·13-s + 0.652·17-s + 1.70·19-s + 1.45·23-s + 0.200·25-s + 1.84·29-s + 0.0491·31-s − 0.553·35-s − 1.79·37-s − 0.742·41-s − 0.423·43-s − 0.697·47-s + 0.531·49-s + 1.03·53-s − 0.311·55-s − 0.961·59-s + 0.281·61-s − 0.397·65-s − 1.41·67-s − 0.557·71-s + 0.459·73-s + 0.862·77-s + 0.708·79-s − 0.572·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 7 | \( 1 + 160.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 279.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 541.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 777.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.68e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 262.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.10e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.19e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.92e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.22e4T + 8.58e9T^{2} \) |
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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
−9.455903034319515240214218579659, −8.426173172738879375533969866516, −7.27084645810238888088950578660, −6.71388505001524908115487013080, −5.51756950911326741555548756914, −4.90981330265932027190485994066, −3.24446857982632755795145761584, −2.80952349191125992328951971780, −1.21817207031493017136964693546, 0,
1.21817207031493017136964693546, 2.80952349191125992328951971780, 3.24446857982632755795145761584, 4.90981330265932027190485994066, 5.51756950911326741555548756914, 6.71388505001524908115487013080, 7.27084645810238888088950578660, 8.426173172738879375533969866516, 9.455903034319515240214218579659
