| L(s) = 1 | + 2·3-s + 3·9-s + 4·13-s − 12·17-s − 12·23-s − 25-s + 4·27-s − 12·29-s + 8·39-s − 16·43-s + 14·49-s − 24·51-s − 24·53-s + 20·61-s − 24·69-s − 2·75-s + 16·79-s + 5·81-s − 24·87-s + 36·101-s − 28·103-s − 36·113-s + 12·117-s − 14·121-s + 127-s − 32·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.10·13-s − 2.91·17-s − 2.50·23-s − 1/5·25-s + 0.769·27-s − 2.22·29-s + 1.28·39-s − 2.43·43-s + 2·49-s − 3.36·51-s − 3.29·53-s + 2.56·61-s − 2.88·69-s − 0.230·75-s + 1.80·79-s + 5/9·81-s − 2.57·87-s + 3.58·101-s − 2.75·103-s − 3.38·113-s + 1.10·117-s − 1.27·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375845554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375845554\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.982222153148387092033322546628, −8.469263684178244302118647800504, −8.145538565376262877494293307202, −7.903974659175388777672978526198, −7.58024952690085604738972559796, −6.84532373938095313448231364234, −6.75257038094241341207078268566, −6.26224197263543577375439853669, −6.09223474528212172014317506852, −5.30028400791078410587480469842, −5.10306258939187834728861438263, −4.32769803377321970550341827360, −4.12390869295632286299296211492, −3.67629698055292085328721798651, −3.57165036542664655042811589298, −2.67263551093330444896113620250, −2.29218140348950653629071844690, −1.76461761232381747837026613504, −1.67794878123580605172989004577, −0.30545434561051649273367858250,
0.30545434561051649273367858250, 1.67794878123580605172989004577, 1.76461761232381747837026613504, 2.29218140348950653629071844690, 2.67263551093330444896113620250, 3.57165036542664655042811589298, 3.67629698055292085328721798651, 4.12390869295632286299296211492, 4.32769803377321970550341827360, 5.10306258939187834728861438263, 5.30028400791078410587480469842, 6.09223474528212172014317506852, 6.26224197263543577375439853669, 6.75257038094241341207078268566, 6.84532373938095313448231364234, 7.58024952690085604738972559796, 7.903974659175388777672978526198, 8.145538565376262877494293307202, 8.469263684178244302118647800504, 8.982222153148387092033322546628
