| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 3·9-s − 11-s − 2·13-s + 4·15-s + 3·17-s − 2·19-s − 2·21-s − 7·23-s + 3·25-s − 4·27-s − 4·29-s + 2·31-s + 2·33-s − 2·35-s + 5·37-s + 4·39-s + 3·41-s − 8·43-s − 6·45-s − 14·47-s − 5·49-s − 6·51-s − 13·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s − 0.301·11-s − 0.554·13-s + 1.03·15-s + 0.727·17-s − 0.458·19-s − 0.436·21-s − 1.45·23-s + 3/5·25-s − 0.769·27-s − 0.742·29-s + 0.359·31-s + 0.348·33-s − 0.338·35-s + 0.821·37-s + 0.640·39-s + 0.468·41-s − 1.21·43-s − 0.894·45-s − 2.04·47-s − 5/7·49-s − 0.840·51-s − 1.78·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 50 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 72 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 110 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 140 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 17 T + 206 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T - 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 216 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−9.373963810743078125124723959727, −8.848852898568488349577971421141, −8.126499495718812081745866907592, −8.055518352651309420271159977492, −7.59937152563170623994539192669, −7.43213314193352180299668055362, −6.58669879134925487042052217497, −6.52498003320454700864163080915, −5.85668119318307291103580531198, −5.68488283963036633335540257132, −4.92770220266544324917678742473, −4.69727667181279747224960386708, −4.34970665648818563445137087396, −3.82172471990144146677918723017, −3.14945507832855181754788707672, −2.77826021906154836928875457247, −1.64753010609905194420961119874, −1.49639559463699724088469444879, 0, 0,
1.49639559463699724088469444879, 1.64753010609905194420961119874, 2.77826021906154836928875457247, 3.14945507832855181754788707672, 3.82172471990144146677918723017, 4.34970665648818563445137087396, 4.69727667181279747224960386708, 4.92770220266544324917678742473, 5.68488283963036633335540257132, 5.85668119318307291103580531198, 6.52498003320454700864163080915, 6.58669879134925487042052217497, 7.43213314193352180299668055362, 7.59937152563170623994539192669, 8.055518352651309420271159977492, 8.126499495718812081745866907592, 8.848852898568488349577971421141, 9.373963810743078125124723959727
