L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 2·5-s + 2·6-s − 7-s − 4·8-s − 4·10-s − 3·11-s − 3·12-s + 7·13-s + 2·14-s − 2·15-s + 5·16-s + 3·17-s − 7·19-s + 6·20-s + 21-s + 6·22-s + 4·24-s + 3·25-s − 14·26-s − 2·27-s − 3·28-s + 6·29-s + 4·30-s + 7·31-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s − 1.41·8-s − 1.26·10-s − 0.904·11-s − 0.866·12-s + 1.94·13-s + 0.534·14-s − 0.516·15-s + 5/4·16-s + 0.727·17-s − 1.60·19-s + 1.34·20-s + 0.218·21-s + 1.27·22-s + 0.816·24-s + 3/5·25-s − 2.74·26-s − 0.384·27-s − 0.566·28-s + 1.11·29-s + 0.730·30-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27984100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | | \( 1 \) |
good | 3 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 27 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 154 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 178 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 97 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−8.055654728816263958943295910872, −7.983632026711558776898517608607, −7.38285736491745071528538787897, −6.77682055724084887237329671925, −6.41165067596489630260515104553, −6.39207443516318198282613810262, −5.95634571640188433173166948228, −5.93741009904395761926143946770, −4.98346214175879970956748007345, −4.98335620458373946661440230188, −4.43479227425195452680727738060, −3.76114212618819706489735497142, −3.26541588307983695276272334365, −2.86840445691376213680526134083, −2.65269785413680882542054702615, −1.78526885550021860631648280189, −1.40621818432775280891407172359, −1.25733118263330262111971044933, 0, 0,
1.25733118263330262111971044933, 1.40621818432775280891407172359, 1.78526885550021860631648280189, 2.65269785413680882542054702615, 2.86840445691376213680526134083, 3.26541588307983695276272334365, 3.76114212618819706489735497142, 4.43479227425195452680727738060, 4.98335620458373946661440230188, 4.98346214175879970956748007345, 5.93741009904395761926143946770, 5.95634571640188433173166948228, 6.39207443516318198282613810262, 6.41165067596489630260515104553, 6.77682055724084887237329671925, 7.38285736491745071528538787897, 7.983632026711558776898517608607, 8.055654728816263958943295910872