| L(s) = 1 | + 3-s + 6·5-s + 2·7-s + 6·11-s − 7·13-s + 6·15-s + 3·17-s + 2·19-s + 2·21-s − 6·23-s + 17·25-s − 27-s − 3·29-s + 8·31-s + 6·33-s + 12·35-s + 7·37-s − 7·39-s + 3·41-s − 10·43-s − 12·47-s + 7·49-s + 3·51-s + 6·53-s + 36·55-s + 2·57-s + 7·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 2.68·5-s + 0.755·7-s + 1.80·11-s − 1.94·13-s + 1.54·15-s + 0.727·17-s + 0.458·19-s + 0.436·21-s − 1.25·23-s + 17/5·25-s − 0.192·27-s − 0.557·29-s + 1.43·31-s + 1.04·33-s + 2.02·35-s + 1.15·37-s − 1.12·39-s + 0.468·41-s − 1.52·43-s − 1.75·47-s + 49-s + 0.420·51-s + 0.824·53-s + 4.85·55-s + 0.264·57-s + 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.465789581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.465789581\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.39630604373407081910258869264, −10.28640645628495921856398236351, −9.716887311965313403743232045366, −9.568997773695613993861017676474, −9.424154880918457675126148905083, −8.823672827469495560122281666167, −8.058424623744113190099056541143, −8.050007403862835446128800774610, −7.06523591181107522302721005016, −6.87411254106722560239472779653, −6.24720655627968707306856005175, −5.74553235032811395336553220431, −5.56980462981863390959685445144, −4.84463068511516099949861750980, −4.43442685339367085143955114496, −3.69194797962191056901510713480, −2.71126122125106236736270124767, −2.46571490286286266500513606038, −1.65603438242717813851675669160, −1.41009298425134880342500189697,
1.41009298425134880342500189697, 1.65603438242717813851675669160, 2.46571490286286266500513606038, 2.71126122125106236736270124767, 3.69194797962191056901510713480, 4.43442685339367085143955114496, 4.84463068511516099949861750980, 5.56980462981863390959685445144, 5.74553235032811395336553220431, 6.24720655627968707306856005175, 6.87411254106722560239472779653, 7.06523591181107522302721005016, 8.050007403862835446128800774610, 8.058424623744113190099056541143, 8.823672827469495560122281666167, 9.424154880918457675126148905083, 9.568997773695613993861017676474, 9.716887311965313403743232045366, 10.28640645628495921856398236351, 10.39630604373407081910258869264
