| L(s) = 1 | − 14·19-s + 7·25-s + 12·29-s − 10·31-s − 10·37-s + 6·47-s + 18·53-s + 18·59-s + 24·83-s + 2·103-s + 22·109-s − 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.21·19-s + 7/5·25-s + 2.22·29-s − 1.79·31-s − 1.64·37-s + 0.875·47-s + 2.47·53-s + 2.34·59-s + 2.63·83-s + 0.197·103-s + 2.10·109-s − 1.12·113-s + 1.72·121-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.667960695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.667960695\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 107 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.028428612603451586522287380622, −8.026475907570667547216724708315, −7.19571912842469547367719109107, −7.00866407414404270157804879262, −6.75848535327920512346273789894, −6.50369004015251591324545535905, −6.04315310811076425445779523108, −5.70676722632084493973262372871, −5.13120895994906970240047178107, −5.05655078076346541028736057769, −4.47761594457905485660716018121, −4.19005759470164429182672936952, −3.72429222487841975771294925665, −3.57254945916163737161655319805, −2.69399820449790123844939848719, −2.60375618833385925078606604664, −1.95241034889330119870062566236, −1.81435211031509406449427572549, −0.797655410376273822697492592943, −0.51862816290019674256169529816,
0.51862816290019674256169529816, 0.797655410376273822697492592943, 1.81435211031509406449427572549, 1.95241034889330119870062566236, 2.60375618833385925078606604664, 2.69399820449790123844939848719, 3.57254945916163737161655319805, 3.72429222487841975771294925665, 4.19005759470164429182672936952, 4.47761594457905485660716018121, 5.05655078076346541028736057769, 5.13120895994906970240047178107, 5.70676722632084493973262372871, 6.04315310811076425445779523108, 6.50369004015251591324545535905, 6.75848535327920512346273789894, 7.00866407414404270157804879262, 7.19571912842469547367719109107, 8.026475907570667547216724708315, 8.028428612603451586522287380622
