| L(s) = 1 | + 4·5-s + 3·9-s − 6·11-s − 16·19-s + 5·25-s + 4·29-s + 4·31-s − 24·41-s + 12·45-s − 24·55-s − 20·59-s − 14·79-s + 9·81-s + 16·89-s − 64·95-s − 18·99-s + 24·101-s − 14·109-s + 31·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 9-s − 1.80·11-s − 3.67·19-s + 25-s + 0.742·29-s + 0.718·31-s − 3.74·41-s + 1.78·45-s − 3.23·55-s − 2.60·59-s − 1.57·79-s + 81-s + 1.69·89-s − 6.56·95-s − 1.80·99-s + 2.38·101-s − 1.34·109-s + 2.81·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01644324955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01644324955\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 9 T^{2} - 208 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 27 T^{2} - 1480 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 185 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.21250933477752441130218541781, −6.69441759163130989770826021464, −6.58551052585754380153113085522, −6.45492950259355283937012833394, −6.41983240329714525238744977530, −6.05481972777082024739420643487, −5.94043774516656480927488681283, −5.51666822624580482865690215764, −5.32457013949904381309749558053, −5.14367182259279865371453991940, −4.70877252135584156490909615219, −4.65023441726917074162145194064, −4.61404768714028892667181800824, −4.18414552022219464378180601435, −3.84026228913242950129517929817, −3.53160333183484700010917107419, −3.25280095826607264231692118461, −2.84909332267963434753625479172, −2.63677986777861922652480865470, −2.21350996757624950219245770409, −2.13953807299700065005759460834, −1.69281579681639743066495187430, −1.68738957364225338642736906179, −1.05431700983908253886378584712, −0.02671729847966380313168690922,
0.02671729847966380313168690922, 1.05431700983908253886378584712, 1.68738957364225338642736906179, 1.69281579681639743066495187430, 2.13953807299700065005759460834, 2.21350996757624950219245770409, 2.63677986777861922652480865470, 2.84909332267963434753625479172, 3.25280095826607264231692118461, 3.53160333183484700010917107419, 3.84026228913242950129517929817, 4.18414552022219464378180601435, 4.61404768714028892667181800824, 4.65023441726917074162145194064, 4.70877252135584156490909615219, 5.14367182259279865371453991940, 5.32457013949904381309749558053, 5.51666822624580482865690215764, 5.94043774516656480927488681283, 6.05481972777082024739420643487, 6.41983240329714525238744977530, 6.45492950259355283937012833394, 6.58551052585754380153113085522, 6.69441759163130989770826021464, 7.21250933477752441130218541781
