| L(s) = 1 | + 6·7-s − 8·11-s + 14·13-s + 86·19-s − 128·23-s − 344·29-s − 158·31-s + 36·37-s + 244·41-s + 390·43-s − 756·47-s + 65·49-s − 268·53-s − 4·59-s − 1.03e3·61-s + 1.62e3·67-s + 276·71-s + 644·73-s − 48·77-s − 944·79-s + 484·83-s − 2.22e3·89-s + 84·91-s + 510·97-s + 844·101-s + 544·103-s − 3.22e3·107-s + ⋯ |
| L(s) = 1 | + 0.323·7-s − 0.219·11-s + 0.298·13-s + 1.03·19-s − 1.16·23-s − 2.20·29-s − 0.915·31-s + 0.159·37-s + 0.929·41-s + 1.38·43-s − 2.34·47-s + 0.189·49-s − 0.694·53-s − 0.00882·59-s − 2.17·61-s + 2.95·67-s + 0.461·71-s + 1.03·73-s − 0.0710·77-s − 1.34·79-s + 0.640·83-s − 2.64·89-s + 0.0967·91-s + 0.533·97-s + 0.831·101-s + 0.520·103-s − 2.90·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 6 T - 29 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 8 T + 1954 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 14 T + 119 p T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9102 T^{2} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 86 T + 9051 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 128 T + 21914 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 344 T + 2078 p T^{2} + 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 158 T + 30347 T^{2} + 158 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 36 T + 75566 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 244 T + 117250 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 390 T + 161563 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 756 T + 338946 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 268 T + 280234 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 364426 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1034 T + 674915 T^{2} + 1034 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1622 T + 1200603 T^{2} - 1622 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 276 T - 53570 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 644 T + 809318 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 944 T + 919262 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 484 T + 460762 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2224 T + 2634898 T^{2} + 2224 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 510 T + 1148995 T^{2} - 510 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.659200095688868914838211273992, −8.327845637601000965541646538534, −7.74596938661892142430171237271, −7.63271301677198280093620835408, −7.36845267733074308784211305340, −6.65650270458210145966859831172, −6.32072133532544498638575214906, −5.85651619189362015616890196507, −5.43198688167205370744747466064, −5.18899958515447941859443083447, −4.60974227012599161546700276652, −4.06942302483626015670741981293, −3.59817969884900890684348407779, −3.41763886745951370487965166031, −2.48761835565738179474531515071, −2.26509670378497213469723037291, −1.43356289792186708397806462179, −1.21511692913021293780627915718, 0, 0,
1.21511692913021293780627915718, 1.43356289792186708397806462179, 2.26509670378497213469723037291, 2.48761835565738179474531515071, 3.41763886745951370487965166031, 3.59817969884900890684348407779, 4.06942302483626015670741981293, 4.60974227012599161546700276652, 5.18899958515447941859443083447, 5.43198688167205370744747466064, 5.85651619189362015616890196507, 6.32072133532544498638575214906, 6.65650270458210145966859831172, 7.36845267733074308784211305340, 7.63271301677198280093620835408, 7.74596938661892142430171237271, 8.327845637601000965541646538534, 8.659200095688868914838211273992
