| L(s) = 1 | + 4-s − 8·5-s + 6·11-s − 2·19-s − 8·20-s + 38·25-s − 4·29-s + 16·31-s + 20·41-s + 6·44-s + 11·49-s − 48·55-s − 64-s + 24·71-s − 2·76-s + 56·79-s + 2·89-s + 16·95-s + 38·100-s + 20·101-s − 40·109-s − 4·116-s + 31·121-s + 16·124-s − 136·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.57·5-s + 1.80·11-s − 0.458·19-s − 1.78·20-s + 38/5·25-s − 0.742·29-s + 2.87·31-s + 3.12·41-s + 0.904·44-s + 11/7·49-s − 6.47·55-s − 1/8·64-s + 2.84·71-s − 0.229·76-s + 6.30·79-s + 0.211·89-s + 1.64·95-s + 19/5·100-s + 1.99·101-s − 3.83·109-s − 0.371·116-s + 2.81·121-s + 1.43·124-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.362454735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.362454735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 7 T^{2} - 1320 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 98 T^{2} + 5115 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12652195725022652012681635679, −6.62763746644612697467267060714, −6.53131881336942963762128828020, −6.46399640729519098331371481951, −6.27998069465095788168720440171, −6.23706117318898479616836317049, −5.58462401351470935066105690735, −5.30683172389033585176192829307, −5.13554359365179694627842582842, −4.67462734962140822993081038951, −4.60047561240050186407363785929, −4.57858527793256774750620682696, −3.99611518415991441354285706033, −3.84496497324274422957138265796, −3.77599288584934984354922882289, −3.70606274175296094692901509450, −3.47805302927770762332887806082, −2.77948182022621543077433377610, −2.76759698276698226191310154036, −2.36387994421114574851446410638, −2.25411336226817456741038637983, −1.43186734595603856876908669797, −0.994186858313060608139304908980, −0.801951463057062804632151637621, −0.51233993741647109054671438546,
0.51233993741647109054671438546, 0.801951463057062804632151637621, 0.994186858313060608139304908980, 1.43186734595603856876908669797, 2.25411336226817456741038637983, 2.36387994421114574851446410638, 2.76759698276698226191310154036, 2.77948182022621543077433377610, 3.47805302927770762332887806082, 3.70606274175296094692901509450, 3.77599288584934984354922882289, 3.84496497324274422957138265796, 3.99611518415991441354285706033, 4.57858527793256774750620682696, 4.60047561240050186407363785929, 4.67462734962140822993081038951, 5.13554359365179694627842582842, 5.30683172389033585176192829307, 5.58462401351470935066105690735, 6.23706117318898479616836317049, 6.27998069465095788168720440171, 6.46399640729519098331371481951, 6.53131881336942963762128828020, 6.62763746644612697467267060714, 7.12652195725022652012681635679
