| L(s) = 1 | − 2·4-s − 10·7-s + 18·13-s + 4·16-s − 42·19-s + 20·28-s + 80·31-s + 50·37-s + 128·43-s − 23·49-s − 36·52-s − 194·61-s − 8·64-s + 262·67-s − 34·73-s + 84·76-s + 234·79-s − 180·91-s + 82·97-s − 26·103-s − 16·109-s − 40·112-s + 240·121-s − 160·124-s + 127-s + 131-s + 420·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.42·7-s + 1.38·13-s + 1/4·16-s − 2.21·19-s + 5/7·28-s + 2.58·31-s + 1.35·37-s + 2.97·43-s − 0.469·49-s − 0.692·52-s − 3.18·61-s − 1/8·64-s + 3.91·67-s − 0.465·73-s + 1.10·76-s + 2.96·79-s − 1.97·91-s + 0.845·97-s − 0.252·103-s − 0.146·109-s − 0.357·112-s + 1.98·121-s − 1.29·124-s + 0.00787·127-s + 0.00763·131-s + 3.15·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.044595330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.044595330\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 240 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 9 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 450 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 21 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1056 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 624 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 64 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3906 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 1230 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 2144 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 117 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10416 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 5790 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 41 T + p^{2} T^{2} )^{2} \) |
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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
−9.466545068184599921199674824625, −9.347961007809038736370674560542, −8.843336536976711612009453967534, −8.435046346267932872607269442227, −7.976143456787218784838568401700, −7.88768964287094247262380070575, −7.05338880430828772794540463433, −6.42977018471664455354639522603, −6.37362001659865149908092510101, −6.11669022322999853965129551991, −5.65713155134435834754461036828, −4.70466518296137287597086681587, −4.58525772092427123404811728486, −3.94980913037106837859697732271, −3.68754695833858244509889452591, −2.98103994201911336432867238980, −2.58809173768703057628543248798, −1.93203843769617748910126047035, −0.886912598145638634744487193349, −0.54737041543364582378672122098,
0.54737041543364582378672122098, 0.886912598145638634744487193349, 1.93203843769617748910126047035, 2.58809173768703057628543248798, 2.98103994201911336432867238980, 3.68754695833858244509889452591, 3.94980913037106837859697732271, 4.58525772092427123404811728486, 4.70466518296137287597086681587, 5.65713155134435834754461036828, 6.11669022322999853965129551991, 6.37362001659865149908092510101, 6.42977018471664455354639522603, 7.05338880430828772794540463433, 7.88768964287094247262380070575, 7.976143456787218784838568401700, 8.435046346267932872607269442227, 8.843336536976711612009453967534, 9.347961007809038736370674560542, 9.466545068184599921199674824625
