| L(s) = 1 | + 2·3-s − 2·4-s + 2·5-s − 9-s − 6·11-s − 4·12-s + 6·13-s + 4·15-s − 2·17-s + 12·19-s − 4·20-s + 12·23-s + 3·25-s − 6·27-s − 6·29-s + 12·31-s − 12·33-s + 2·36-s − 4·37-s + 12·39-s − 4·41-s + 4·43-s + 12·44-s − 2·45-s − 6·47-s − 4·51-s − 12·52-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 0.894·5-s − 1/3·9-s − 1.80·11-s − 1.15·12-s + 1.66·13-s + 1.03·15-s − 0.485·17-s + 2.75·19-s − 0.894·20-s + 2.50·23-s + 3/5·25-s − 1.15·27-s − 1.11·29-s + 2.15·31-s − 2.08·33-s + 1/3·36-s − 0.657·37-s + 1.92·39-s − 0.624·41-s + 0.609·43-s + 1.80·44-s − 0.298·45-s − 0.875·47-s − 0.560·51-s − 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.824195009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824195009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 104 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 135 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 257 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−12.65123854408679236398752966025, −11.73477395593774816396878547392, −11.23727897805168480757040205306, −11.00202302594167867959362115395, −10.03940226125010060606135121765, −10.02444638413333487836632778087, −9.154333733674727999064963391188, −9.039973241264424606300629506558, −8.411833973854258661558160498613, −8.336620381965841247969827965325, −7.37480225697226856105903711973, −7.13974971949884308544174915442, −6.01837265074468781325981483010, −5.65488311335572233497147904842, −5.00788806556701209132586036557, −4.69114213608509091054076420224, −3.28226507776473392443362812337, −3.22276589798217675804356448983, −2.50776347176226660734513924176, −1.16360101941472512386216658081,
1.16360101941472512386216658081, 2.50776347176226660734513924176, 3.22276589798217675804356448983, 3.28226507776473392443362812337, 4.69114213608509091054076420224, 5.00788806556701209132586036557, 5.65488311335572233497147904842, 6.01837265074468781325981483010, 7.13974971949884308544174915442, 7.37480225697226856105903711973, 8.336620381965841247969827965325, 8.411833973854258661558160498613, 9.039973241264424606300629506558, 9.154333733674727999064963391188, 10.02444638413333487836632778087, 10.03940226125010060606135121765, 11.00202302594167867959362115395, 11.23727897805168480757040205306, 11.73477395593774816396878547392, 12.65123854408679236398752966025
