| L(s) = 1 | + 4·4-s + 12·16-s − 2·25-s + 40·31-s − 20·49-s + 32·64-s − 40·79-s − 8·100-s + 20·121-s + 160·124-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 − 80·196-s + 197-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s − 2/5·25-s + 7.18·31-s − 2.85·49-s + 4·64-s − 4.50·79-s − 4/5·100-s + 1.81·121-s + 14.3·124-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 − 5.71·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.424396302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.424396302\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.233946935494726563831081087663, −8.152683965029034971226494637639, −7.78643897706455061667956405243, −7.54611471468978967605890257023, −7.34110727449173129191484639931, −6.84312557758612958948445690099, −6.73474638320270043973970995103, −6.64361071551697573271240010768, −6.31954583954699879361530998778, −5.92099877812555013315550642710, −5.90622160224933358379671670872, −5.83682600018824827551550231437, −4.95529512687216041590776436676, −4.94000032488637973811100299843, −4.59890365192795359426966641373, −4.40062229500986507554006507399, −3.97570021832298117087896991909, −3.40291388318671117170332009376, −3.21382196039481501191440307182, −2.72121188527205865344219831692, −2.71053118871072144690534252891, −2.44764899536311490695836001089, −1.70787606205421537001123516441, −1.33458986035344808092351792073, −0.932609320217043395862093561078,
0.932609320217043395862093561078, 1.33458986035344808092351792073, 1.70787606205421537001123516441, 2.44764899536311490695836001089, 2.71053118871072144690534252891, 2.72121188527205865344219831692, 3.21382196039481501191440307182, 3.40291388318671117170332009376, 3.97570021832298117087896991909, 4.40062229500986507554006507399, 4.59890365192795359426966641373, 4.94000032488637973811100299843, 4.95529512687216041590776436676, 5.83682600018824827551550231437, 5.90622160224933358379671670872, 5.92099877812555013315550642710, 6.31954583954699879361530998778, 6.64361071551697573271240010768, 6.73474638320270043973970995103, 6.84312557758612958948445690099, 7.34110727449173129191484639931, 7.54611471468978967605890257023, 7.78643897706455061667956405243, 8.152683965029034971226494637639, 8.233946935494726563831081087663
