| L(s) = 1 | − 4·4-s + 12·16-s + 92·25-s + 256·37-s + 176·43-s − 32·64-s + 480·67-s + 368·79-s − 368·100-s + 280·109-s + 468·121-s + 127-s + 131-s + 137-s + 139-s − 1.02e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 352·169-s − 704·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s + 3.67·25-s + 6.91·37-s + 4.09·43-s − 1/2·64-s + 7.16·67-s + 4.65·79-s − 3.67·100-s + 2.56·109-s + 3.86·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 6.91·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.08·169-s − 4.09·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.602708304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.602708304\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 234 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 176 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 690 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 1050 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 432 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 770 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 64 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 2962 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 206 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5280 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3038 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 4704 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 10010 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 5040 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 92 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 1234 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15442 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 18096 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.34551252542406505638621817932, −6.53270812195965430283299446912, −6.51765581789409209756598516428, −6.49135433342551387115967289084, −6.26656363743895845733921610655, −5.85892779683274785563746997203, −5.74478191134501118672409423052, −5.36074127679982741914115330568, −5.15494788372360179783538372542, −4.79013568905701027944009123222, −4.77635180840864639505962931359, −4.51373838564450187056943290538, −4.26405032177997103244180749411, −3.92333577198126412740137709391, −3.75923726436234364632360657169, −3.48434341800496029820808240195, −3.11662978885447902695817070140, −2.60140568911097692391326393597, −2.47627188026566385276826774656, −2.43483088063024155219764390066, −2.10069955843459331158687924849, −0.983934921236531911442892866607, −0.847246425636990681161218393380, −0.805067473172015488345116749782, −0.792999614294037286584586582458,
0.792999614294037286584586582458, 0.805067473172015488345116749782, 0.847246425636990681161218393380, 0.983934921236531911442892866607, 2.10069955843459331158687924849, 2.43483088063024155219764390066, 2.47627188026566385276826774656, 2.60140568911097692391326393597, 3.11662978885447902695817070140, 3.48434341800496029820808240195, 3.75923726436234364632360657169, 3.92333577198126412740137709391, 4.26405032177997103244180749411, 4.51373838564450187056943290538, 4.77635180840864639505962931359, 4.79013568905701027944009123222, 5.15494788372360179783538372542, 5.36074127679982741914115330568, 5.74478191134501118672409423052, 5.85892779683274785563746997203, 6.26656363743895845733921610655, 6.49135433342551387115967289084, 6.51765581789409209756598516428, 6.53270812195965430283299446912, 7.34551252542406505638621817932
