| L(s) = 1 | − 20·25-s + 64·67-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 4·25-s + 7.81·67-s − 4·121-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 + 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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.348807697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.348807697\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 194 T^{4} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 2062 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^3$ | \( 1 - 1966 T^{4} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 7682 T^{4} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.56268289066454963226288484590, −5.17070303823346492077175950706, −5.05121838277036801930042013401, −4.92487081931468815963232915729, −4.87411078232184658234922266199, −4.26476745704602860369204133845, −4.19353513240785140615006454540, −4.14053414889633171271156859426, −4.04111241080636171150165958289, −3.59818772678345500450528097405, −3.58518650484385927523282139172, −3.50399956144488211330101940030, −3.36689300676674853445385804829, −2.77819491098741114169580916619, −2.62313114637905072766341270248, −2.49961923760053331634484074454, −2.42505165566482354671829267791, −1.93481572947072796841943348901, −1.85888278633763836280341812485, −1.73589394875469425226431388105, −1.55519748223773179259816827057, −1.01251169998746553677197041727, −0.67716759983748236729007642184, −0.56507748145586868443495279768, −0.29900260853018154502507582581,
0.29900260853018154502507582581, 0.56507748145586868443495279768, 0.67716759983748236729007642184, 1.01251169998746553677197041727, 1.55519748223773179259816827057, 1.73589394875469425226431388105, 1.85888278633763836280341812485, 1.93481572947072796841943348901, 2.42505165566482354671829267791, 2.49961923760053331634484074454, 2.62313114637905072766341270248, 2.77819491098741114169580916619, 3.36689300676674853445385804829, 3.50399956144488211330101940030, 3.58518650484385927523282139172, 3.59818772678345500450528097405, 4.04111241080636171150165958289, 4.14053414889633171271156859426, 4.19353513240785140615006454540, 4.26476745704602860369204133845, 4.87411078232184658234922266199, 4.92487081931468815963232915729, 5.05121838277036801930042013401, 5.17070303823346492077175950706, 5.56268289066454963226288484590
