| L(s) = 1 | + 3-s + 2·4-s + 4·5-s + 3·7-s − 11-s + 2·12-s − 5·13-s + 4·15-s − 2·17-s − 4·19-s + 8·20-s + 3·21-s + 4·23-s + 2·25-s − 27-s + 6·28-s + 10·31-s − 33-s + 12·35-s + 6·37-s − 5·39-s + 8·41-s − 5·43-s − 2·44-s + 7·49-s − 2·51-s − 10·52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 4-s + 1.78·5-s + 1.13·7-s − 0.301·11-s + 0.577·12-s − 1.38·13-s + 1.03·15-s − 0.485·17-s − 0.917·19-s + 1.78·20-s + 0.654·21-s + 0.834·23-s + 2/5·25-s − 0.192·27-s + 1.13·28-s + 1.79·31-s − 0.174·33-s + 2.02·35-s + 0.986·37-s − 0.800·39-s + 1.24·41-s − 0.762·43-s − 0.301·44-s + 49-s − 0.280·51-s − 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.746972264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.746972264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−11.56255950723112049947592506413, −10.81678044773936391783532693265, −10.36546363481075315041401481862, −10.09843550297241741908486596020, −9.736898680263919613453495749469, −9.086959780318959595116549245163, −8.660869948825948615679052357405, −8.350052317819501847877473493045, −7.45289854872019843962536436685, −7.39087660368145928505961148274, −6.73397557999703182995456990489, −6.20740050845911403855201351866, −5.64539120740918972859028818568, −5.37661663680503849112181035886, −4.39589301572929641217737349104, −4.34252460016639136631536518347, −2.73462605109970399686229387303, −2.55122776933982552135848798009, −2.16678366987858238140260750155, −1.39720427620491547902621904285,
1.39720427620491547902621904285, 2.16678366987858238140260750155, 2.55122776933982552135848798009, 2.73462605109970399686229387303, 4.34252460016639136631536518347, 4.39589301572929641217737349104, 5.37661663680503849112181035886, 5.64539120740918972859028818568, 6.20740050845911403855201351866, 6.73397557999703182995456990489, 7.39087660368145928505961148274, 7.45289854872019843962536436685, 8.350052317819501847877473493045, 8.660869948825948615679052357405, 9.086959780318959595116549245163, 9.736898680263919613453495749469, 10.09843550297241741908486596020, 10.36546363481075315041401481862, 10.81678044773936391783532693265, 11.56255950723112049947592506413
