L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 6·11-s + 2·13-s + 5·16-s + 6·17-s + 2·19-s + 12·22-s − 6·23-s − 7·25-s − 4·26-s − 6·29-s + 2·31-s − 6·32-s − 12·34-s + 4·37-s − 4·38-s + 12·41-s + 4·43-s − 18·44-s + 12·46-s − 6·47-s + 14·50-s + 6·52-s − 12·53-s + 12·58-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1.80·11-s + 0.554·13-s + 5/4·16-s + 1.45·17-s + 0.458·19-s + 2.55·22-s − 1.25·23-s − 7/5·25-s − 0.784·26-s − 1.11·29-s + 0.359·31-s − 1.06·32-s − 2.05·34-s + 0.657·37-s − 0.648·38-s + 1.87·41-s + 0.609·43-s − 2.71·44-s + 1.76·46-s − 0.875·47-s + 1.97·50-s + 0.832·52-s − 1.64·53-s + 1.57·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 132 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 247 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−7.74671721865724625637901407200, −7.63746402065382400153937252411, −7.32477436173228081224845685000, −6.68821685910079755402141530246, −6.22688449808974176248504567909, −5.95704212229321978482626246225, −5.65548500378443992368949053229, −5.59654489521095325153702553735, −4.75771356584939530616134999571, −4.68694700754188630326622835804, −3.90075934228242579549846329763, −3.59592596902147513212601137403, −3.19988105402462027607504491121, −2.77747775883350069513971786582, −2.16501366495056473369659127140, −2.15034526022237570402147650190, −1.28374999424732012170339337441, −1.05546918753098166008581357206, 0, 0,
1.05546918753098166008581357206, 1.28374999424732012170339337441, 2.15034526022237570402147650190, 2.16501366495056473369659127140, 2.77747775883350069513971786582, 3.19988105402462027607504491121, 3.59592596902147513212601137403, 3.90075934228242579549846329763, 4.68694700754188630326622835804, 4.75771356584939530616134999571, 5.59654489521095325153702553735, 5.65548500378443992368949053229, 5.95704212229321978482626246225, 6.22688449808974176248504567909, 6.68821685910079755402141530246, 7.32477436173228081224845685000, 7.63746402065382400153937252411, 7.74671721865724625637901407200