| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s − 2·10-s + 5·11-s + 6·12-s − 5·13-s − 4·14-s + 2·15-s + 5·16-s + 17-s − 6·18-s + 6·19-s + 3·20-s + 4·21-s − 10·22-s + 6·23-s − 8·24-s − 8·25-s + 10·26-s + 4·27-s + 6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.447·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 0.632·10-s + 1.50·11-s + 1.73·12-s − 1.38·13-s − 1.06·14-s + 0.516·15-s + 5/4·16-s + 0.242·17-s − 1.41·18-s + 1.37·19-s + 0.670·20-s + 0.872·21-s − 2.13·22-s + 1.25·23-s − 1.63·24-s − 8/5·25-s + 1.96·26-s + 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3261636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3261636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.206724189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.206724189\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 106 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 17 T + 203 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 115 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 130 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 214 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.298433090408731294838024574504, −9.279424331475009235633344523359, −8.665797870414093341501966925817, −8.453075481600496769179304480990, −7.86763275422140105216045181877, −7.66262482412385375091939243922, −7.17302011345114628549993220869, −7.12635878604376108567755247267, −6.28922560627001542241876197496, −6.25197725933288877378846039427, −5.26400423941943234696895049036, −5.20589013382209860034012882323, −4.36520808553268668867170281127, −4.03907016185936146808882001983, −3.33499463359979961047515524454, −2.89597781223468751812472352518, −2.27652940938442076779708456013, −2.06193865823858659526681587563, −1.05807643786077319072737285195, −1.03463634090599930554447748451,
1.03463634090599930554447748451, 1.05807643786077319072737285195, 2.06193865823858659526681587563, 2.27652940938442076779708456013, 2.89597781223468751812472352518, 3.33499463359979961047515524454, 4.03907016185936146808882001983, 4.36520808553268668867170281127, 5.20589013382209860034012882323, 5.26400423941943234696895049036, 6.25197725933288877378846039427, 6.28922560627001542241876197496, 7.12635878604376108567755247267, 7.17302011345114628549993220869, 7.66262482412385375091939243922, 7.86763275422140105216045181877, 8.453075481600496769179304480990, 8.665797870414093341501966925817, 9.279424331475009235633344523359, 9.298433090408731294838024574504
