| L(s) = 1 | + 6·9-s + 4·25-s − 20·49-s − 56·73-s + 27·81-s + 8·97-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 24·225-s + ⋯ |
| L(s) = 1 | + 2·9-s + 4/5·25-s − 2.85·49-s − 6.55·73-s + 3·81-s + 0.812·97-s − 1.81·121-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 + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 8/5·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.073935767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.073935767\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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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
−7.32570669545723150595181355179, −7.16711275172875094805076375514, −7.11533425774731656430088601478, −6.88286165107540979374100367524, −6.37453156957916658313728768998, −6.33069846030775356871813541348, −6.07465737404072177471866593433, −5.76290138329858101247113134895, −5.74305601871394028333282368187, −5.09143092347430032188987251817, −4.89664487839473580130220161652, −4.72500145604183245889614597180, −4.64500702455603336864548345378, −4.41892135810656244908119157414, −3.83292364910677482169371624067, −3.75655952466733903797859199920, −3.64304143512017664889994801086, −3.03813145797375299224413978357, −2.89904344157900136338280407218, −2.54947405039769471395279578509, −2.18484952672149358638869733291, −1.49911110788580926479746368750, −1.42884296640537046417987972835, −1.36425066558286993075187059679, −0.36826834496503402790335027340,
0.36826834496503402790335027340, 1.36425066558286993075187059679, 1.42884296640537046417987972835, 1.49911110788580926479746368750, 2.18484952672149358638869733291, 2.54947405039769471395279578509, 2.89904344157900136338280407218, 3.03813145797375299224413978357, 3.64304143512017664889994801086, 3.75655952466733903797859199920, 3.83292364910677482169371624067, 4.41892135810656244908119157414, 4.64500702455603336864548345378, 4.72500145604183245889614597180, 4.89664487839473580130220161652, 5.09143092347430032188987251817, 5.74305601871394028333282368187, 5.76290138329858101247113134895, 6.07465737404072177471866593433, 6.33069846030775356871813541348, 6.37453156957916658313728768998, 6.88286165107540979374100367524, 7.11533425774731656430088601478, 7.16711275172875094805076375514, 7.32570669545723150595181355179
