| L(s) = 1 | + 4·4-s − 12·9-s − 8·11-s + 12·16-s − 10·25-s − 48·36-s − 32·44-s − 6·49-s + 32·64-s + 90·81-s + 96·99-s − 40·100-s − 4·121-s + 127-s + 131-s + 137-s + 139-s − 144·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s − 96·176-s + 179-s + ⋯ |
| L(s) = 1 | + 2·4-s − 4·9-s − 2.41·11-s + 3·16-s − 2·25-s − 8·36-s − 4.82·44-s − 6/7·49-s + 4·64-s + 10·81-s + 9.64·99-s − 4·100-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 12·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s − 7.23·176-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7748877522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7748877522\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−8.718779477908862890402083894290, −8.149453468914424772475614206002, −7.84575001163635551796609561235, −7.84501915921938188901581207127, −7.83456578543562229550049433579, −7.77574494747414507068019994680, −6.87066805789182194305403868038, −6.71661080634473726664643227949, −6.68437098038172920798169694916, −6.08598336037165351299571389172, −5.91663821221873493431851192087, −5.66719393853226538374610995631, −5.58633281163088331963535669494, −5.39724669473264633519030471226, −5.05661512192263075397131105126, −4.68321640491882346727254862445, −4.05048519308636927800139171088, −3.48059889040351849661163209822, −3.24947152112251740281402331041, −3.14624889166483501802787244592, −2.56121618158740582142065350812, −2.44880743892934855220194940368, −2.31572285969836195690290922794, −1.67728263803887682959896456342, −0.36932729754291788699717431392,
0.36932729754291788699717431392, 1.67728263803887682959896456342, 2.31572285969836195690290922794, 2.44880743892934855220194940368, 2.56121618158740582142065350812, 3.14624889166483501802787244592, 3.24947152112251740281402331041, 3.48059889040351849661163209822, 4.05048519308636927800139171088, 4.68321640491882346727254862445, 5.05661512192263075397131105126, 5.39724669473264633519030471226, 5.58633281163088331963535669494, 5.66719393853226538374610995631, 5.91663821221873493431851192087, 6.08598336037165351299571389172, 6.68437098038172920798169694916, 6.71661080634473726664643227949, 6.87066805789182194305403868038, 7.77574494747414507068019994680, 7.83456578543562229550049433579, 7.84501915921938188901581207127, 7.84575001163635551796609561235, 8.149453468914424772475614206002, 8.718779477908862890402083894290
