| L(s) = 1 | + 3·4-s − 10·11-s + 5·16-s + 4·19-s + 20·29-s + 12·31-s − 20·41-s − 30·44-s + 14·49-s + 10·59-s − 22·61-s + 3·64-s + 10·71-s + 12·76-s − 24·79-s − 4·109-s + 60·116-s + 53·121-s + 36·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 3.01·11-s + 5/4·16-s + 0.917·19-s + 3.71·29-s + 2.15·31-s − 3.12·41-s − 4.52·44-s + 2·49-s + 1.30·59-s − 2.81·61-s + 3/8·64-s + 1.18·71-s + 1.37·76-s − 2.70·79-s − 0.383·109-s + 5.57·116-s + 4.81·121-s + 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.378734185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.378734185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61215149442965429179696178878, −10.29913232160946976238992277696, −10.10786384415216704694864379362, −9.814822359802591843604584831833, −8.709926843696194301552798531265, −8.374525726957246797459728626372, −8.119890996075366631035973334630, −7.68416439140278099668814385769, −7.17293544227318166610320735571, −6.72418497449469969577424433928, −6.45898072218531940599079319514, −5.61048237473385696343641008988, −5.49565002844744699528455313463, −4.66025579059760086623038260439, −4.58968620648555116462124001359, −3.04302569326314745486625824455, −3.02363642255929360321439360863, −2.64428208568049858198327692691, −1.88281778295617797064419698537, −0.804923698608643046881818896052,
0.804923698608643046881818896052, 1.88281778295617797064419698537, 2.64428208568049858198327692691, 3.02363642255929360321439360863, 3.04302569326314745486625824455, 4.58968620648555116462124001359, 4.66025579059760086623038260439, 5.49565002844744699528455313463, 5.61048237473385696343641008988, 6.45898072218531940599079319514, 6.72418497449469969577424433928, 7.17293544227318166610320735571, 7.68416439140278099668814385769, 8.119890996075366631035973334630, 8.374525726957246797459728626372, 8.709926843696194301552798531265, 9.814822359802591843604584831833, 10.10786384415216704694864379362, 10.29913232160946976238992277696, 10.61215149442965429179696178878
