| L(s) = 1 | − 9-s + 12·23-s − 25-s + 12·41-s + 12·47-s − 14·49-s + 24·71-s + 4·73-s + 81-s − 12·89-s + 20·97-s − 24·103-s + 24·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.50·23-s − 1/5·25-s + 1.87·41-s + 1.75·47-s − 2·49-s + 2.84·71-s + 0.468·73-s + 1/9·81-s − 1.27·89-s + 2.03·97-s − 2.36·103-s + 2.25·113-s + 2·121-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 − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.044746517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.044746517\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.16795377436721515360165165108, −9.791404174205329236913538818639, −9.252112663245444521887388804786, −9.149411524924051458065760055421, −8.568646420959776886290290182067, −8.236193129165224611628144615794, −7.63226990999477208812031006376, −7.33051831424741093825018171912, −6.89946925760633565233531571413, −6.35276728339730905888290425773, −6.01164261331180684962665386318, −5.41986070982991468881665365993, −4.88878109577062173430231478200, −4.73310172678691463969111759016, −3.77426431280565518069822193414, −3.58555419156829197477169061376, −2.59629161496953590561914887278, −2.58153010550456342368192956004, −1.42719525854933028490659639719, −0.71968087072549679280678345156,
0.71968087072549679280678345156, 1.42719525854933028490659639719, 2.58153010550456342368192956004, 2.59629161496953590561914887278, 3.58555419156829197477169061376, 3.77426431280565518069822193414, 4.73310172678691463969111759016, 4.88878109577062173430231478200, 5.41986070982991468881665365993, 6.01164261331180684962665386318, 6.35276728339730905888290425773, 6.89946925760633565233531571413, 7.33051831424741093825018171912, 7.63226990999477208812031006376, 8.236193129165224611628144615794, 8.568646420959776886290290182067, 9.149411524924051458065760055421, 9.252112663245444521887388804786, 9.791404174205329236913538818639, 10.16795377436721515360165165108
