| L(s) = 1 | − 4-s − 9-s + 2·11-s + 16-s − 10·19-s + 20·29-s + 4·31-s + 36-s − 6·41-s − 2·44-s + 5·49-s + 10·59-s + 24·61-s − 64-s − 6·71-s + 10·76-s + 30·79-s + 81-s − 2·99-s − 6·101-s − 20·116-s + 3·121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 0.603·11-s + 1/4·16-s − 2.29·19-s + 3.71·29-s + 0.718·31-s + 1/6·36-s − 0.937·41-s − 0.301·44-s + 5/7·49-s + 1.30·59-s + 3.07·61-s − 1/8·64-s − 0.712·71-s + 1.14·76-s + 3.37·79-s + 1/9·81-s − 0.201·99-s − 0.597·101-s − 1.85·116-s + 3/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.896035502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.896035502\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.709863148701918312615903586376, −9.040827536776865768020629037608, −8.538533668758119366666258763509, −8.457478931183849607730687169735, −8.322868756853070966165183921715, −7.73891710252655149277435931695, −6.85465476248640614959694726087, −6.83736774897389098858748792812, −6.33904070843674182344452473442, −6.14855567049655559765034940195, −5.38871748063097559782520478074, −4.95747254270208951105980589045, −4.62574475939430473209109883844, −4.11339592846237972178880562447, −3.79347440972698034962594180443, −3.15736162728139714295843891126, −2.37733769216443385724127723942, −2.30353162485967074721987557855, −1.17113146705776060952018823806, −0.60638565727199238704093605036,
0.60638565727199238704093605036, 1.17113146705776060952018823806, 2.30353162485967074721987557855, 2.37733769216443385724127723942, 3.15736162728139714295843891126, 3.79347440972698034962594180443, 4.11339592846237972178880562447, 4.62574475939430473209109883844, 4.95747254270208951105980589045, 5.38871748063097559782520478074, 6.14855567049655559765034940195, 6.33904070843674182344452473442, 6.83736774897389098858748792812, 6.85465476248640614959694726087, 7.73891710252655149277435931695, 8.322868756853070966165183921715, 8.457478931183849607730687169735, 8.538533668758119366666258763509, 9.040827536776865768020629037608, 9.709863148701918312615903586376
