| L(s) = 1 | + 4-s − 3·9-s − 12·19-s + 7·25-s + 6·31-s − 3·36-s + 4·37-s − 32·43-s − 64-s − 4·67-s − 24·73-s − 12·76-s + 2·79-s + 7·100-s + 12·103-s + 4·109-s − 13·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 9-s − 2.75·19-s + 7/5·25-s + 1.07·31-s − 1/2·36-s + 0.657·37-s − 4.87·43-s − 1/8·64-s − 0.488·67-s − 2.80·73-s − 1.37·76-s + 0.225·79-s + 7/10·100-s + 1.18·103-s + 0.383·109-s − 1.18·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.133887736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133887736\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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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
−8.631785686429059526412987142249, −8.381155258170212290019669483899, −8.112323874867098747610497368273, −7.952239675708179221655396531180, −7.60261531279878153504458801597, −7.17278527525718983402355040025, −6.91568104845687697071114094126, −6.65573610995328430158597105181, −6.52584454738619365605876139162, −6.39250967640094119487775102379, −5.87823666986884176736188306329, −5.78878148030345982863838181468, −5.45410427827062009579743208529, −4.88470369449087313025522077718, −4.73761452183314721066082341022, −4.64672727204623447297349475668, −4.10103047734304976859920472193, −3.91855333372078654525208845638, −3.13556493763568122458555301286, −3.12722002067547892707620702550, −2.89930122969985824905293277113, −2.31818064027532387097558645916, −1.74830994624382194145255407892, −1.70928767573067378324751072299, −0.47417160969388404744112237701,
0.47417160969388404744112237701, 1.70928767573067378324751072299, 1.74830994624382194145255407892, 2.31818064027532387097558645916, 2.89930122969985824905293277113, 3.12722002067547892707620702550, 3.13556493763568122458555301286, 3.91855333372078654525208845638, 4.10103047734304976859920472193, 4.64672727204623447297349475668, 4.73761452183314721066082341022, 4.88470369449087313025522077718, 5.45410427827062009579743208529, 5.78878148030345982863838181468, 5.87823666986884176736188306329, 6.39250967640094119487775102379, 6.52584454738619365605876139162, 6.65573610995328430158597105181, 6.91568104845687697071114094126, 7.17278527525718983402355040025, 7.60261531279878153504458801597, 7.952239675708179221655396531180, 8.112323874867098747610497368273, 8.381155258170212290019669483899, 8.631785686429059526412987142249
