L(s) = 1 | + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.733 + 0.680i)5-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)10-s + (0.826 − 0.563i)11-s + (0.0747 + 0.997i)13-s + (−0.733 − 0.680i)16-s + (0.623 − 0.781i)17-s + 19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)26-s + ⋯ |
L(s) = 1 | + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.733 + 0.680i)5-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)10-s + (0.826 − 0.563i)11-s + (0.0747 + 0.997i)13-s + (−0.733 − 0.680i)16-s + (0.623 − 0.781i)17-s + 19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.676542287 - 1.074382474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.676542287 - 1.074382474i\) |
\(L(1)\) |
\(\approx\) |
\(1.465862105 - 0.5540779126i\) |
\(L(1)\) |
\(\approx\) |
\(1.465862105 - 0.5540779126i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 5 | \( 1 + (-0.733 + 0.680i)T \) |
| 11 | \( 1 + (0.826 - 0.563i)T \) |
| 13 | \( 1 + (0.0747 + 0.997i)T \) |
| 17 | \( 1 + (0.623 - 0.781i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.365 - 0.930i)T \) |
| 29 | \( 1 + (0.365 + 0.930i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.623 - 0.781i)T \) |
| 41 | \( 1 + (-0.733 + 0.680i)T \) |
| 43 | \( 1 + (-0.733 - 0.680i)T \) |
| 47 | \( 1 + (0.826 - 0.563i)T \) |
| 53 | \( 1 + (0.623 + 0.781i)T \) |
| 59 | \( 1 + (0.955 - 0.294i)T \) |
| 61 | \( 1 + (0.365 + 0.930i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.623 + 0.781i)T \) |
| 73 | \( 1 + (-0.900 + 0.433i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.0747 - 0.997i)T \) |
| 89 | \( 1 + (-0.900 + 0.433i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.09314740734504601895391825975, −23.38045253481539644058595759156, −22.689230959333969372249063080821, −21.84503490532467357014402352083, −20.78127010985973345660038615910, −20.12270534128603140529011045372, −19.332433506910202910476228697779, −17.824425707218885894766321910864, −17.10710144320970149156761922898, −16.26259427238768971751504125323, −15.38479189703208265111476017335, −14.82292872133431231323563300772, −13.67180397219347546154720730426, −12.7850723234100963065778903819, −12.0648247943241060764297798939, −11.35618332060132767417941330537, −9.89329121424410951497180231897, −8.657888245756760745562956250881, −7.845889119428439284693573579053, −7.05244972316248989780595589259, −5.78078698809848527468685061642, −4.973776415025015398949312802454, −3.92154144915499406116885901288, −3.15832643637450484032869731156, −1.38826369413143759887406892959,
1.03060564945187868068069480894, 2.53437570385384971068710886668, 3.49078789570785268849981010333, 4.26590522518396230659548334430, 5.4643792430755004005992371117, 6.62902598269171232291292582431, 7.27636963877607529061995846197, 8.80526380469740712160789306375, 9.83095336478668406821820875800, 10.915132300463475098596520807467, 11.65532196241888587532735207025, 12.14883847848917176876129171710, 13.51249396527542935369963850184, 14.30669738752735998984683857927, 14.8236707899146704491866888212, 16.02701070566997992148469459910, 16.61611415770275033249198219744, 18.44539248409374236645883661790, 18.7350736017871442271101652301, 19.78390254085368352583562063068, 20.40002309756522034748658289883, 21.570912903425763193646083026035, 22.18095275299961300144926134472, 22.9481430556792267709556774595, 23.71207045761721615650493789591