| L(s) = 1 | − 4·4-s + 6·11-s + 16·16-s − 64·19-s − 102·29-s + 166·31-s − 216·41-s − 24·44-s + 490·49-s − 24·59-s + 460·61-s − 64·64-s + 240·71-s + 256·76-s + 1.47e3·79-s + 240·89-s + 66·101-s + 1.11e3·109-s + 408·116-s − 2.63e3·121-s − 664·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.164·11-s + 1/4·16-s − 0.772·19-s − 0.653·29-s + 0.961·31-s − 0.822·41-s − 0.0822·44-s + 10/7·49-s − 0.0529·59-s + 0.965·61-s − 1/8·64-s + 0.401·71-s + 0.386·76-s + 2.10·79-s + 0.285·89-s + 0.0650·101-s + 0.977·109-s + 0.326·116-s − 1.97·121-s − 0.480·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.987848921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.987848921\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2185 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8305 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 14533 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 51 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 83 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2710 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 69613 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 74315 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 20342 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 p^{2} T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 445202 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 739 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 35822 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 870818 T^{2} + p^{6} T^{4} \) |
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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
−9.333295974564160485750672812996, −9.081101992182355011682442622514, −8.621364392604750790901629459020, −8.319467675358387956005016514596, −7.81698216259012276788035659997, −7.53996167256523387228459801265, −6.93882309469375426305895910095, −6.42495273315638709617968331908, −6.36163780333701130131262927643, −5.47258492726543146044955171242, −5.39788189416496315386977716878, −4.75296624923259195709509322823, −4.33131595732919936952573316118, −3.82159665785490088951561790468, −3.51465373892718836406268637997, −2.75830274662810522591375361259, −2.27931217844254309294092263558, −1.68672596964360238051469409008, −0.934576430724894554724519768935, −0.39148340628641028495168822801,
0.39148340628641028495168822801, 0.934576430724894554724519768935, 1.68672596964360238051469409008, 2.27931217844254309294092263558, 2.75830274662810522591375361259, 3.51465373892718836406268637997, 3.82159665785490088951561790468, 4.33131595732919936952573316118, 4.75296624923259195709509322823, 5.39788189416496315386977716878, 5.47258492726543146044955171242, 6.36163780333701130131262927643, 6.42495273315638709617968331908, 6.93882309469375426305895910095, 7.53996167256523387228459801265, 7.81698216259012276788035659997, 8.319467675358387956005016514596, 8.621364392604750790901629459020, 9.081101992182355011682442622514, 9.333295974564160485750672812996
