| L(s) = 1 | + 2-s − 2·3-s − 2·6-s + 8-s + 3·9-s − 6·11-s − 2·13-s − 16-s − 2·17-s + 3·18-s − 6·19-s − 6·22-s + 2·23-s − 2·24-s − 2·26-s − 4·27-s − 8·29-s − 6·32-s + 12·33-s − 2·34-s + 6·37-s − 6·38-s + 4·39-s + 12·43-s + 2·46-s − 14·47-s + 2·48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 0.816·6-s + 0.353·8-s + 9-s − 1.80·11-s − 0.554·13-s − 1/4·16-s − 0.485·17-s + 0.707·18-s − 1.37·19-s − 1.27·22-s + 0.417·23-s − 0.408·24-s − 0.392·26-s − 0.769·27-s − 1.48·29-s − 1.06·32-s + 2.08·33-s − 0.342·34-s + 0.986·37-s − 0.973·38-s + 0.640·39-s + 1.82·43-s + 0.294·46-s − 2.04·47-s + 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T - 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24 T + 265 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 206 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.043672229136358257505826123348, −8.023997210289222093489070038720, −7.56581651951199556792667004157, −7.00415047295711269708368840159, −6.97289060689875183716619875266, −6.40676274655494083377340544487, −5.87937415617397616000272474492, −5.79599136724861448291668108596, −5.14549908841918639026426857201, −4.98769286084783640906621018650, −4.69637272779172553927353758102, −4.24134240221525463644226823929, −3.89103525500661360235787154837, −3.36147409196601931801549610534, −2.63039928770751640563098317364, −2.28031633126246932878838666430, −1.92383174007866700534331407499, −1.06079825768261876375865275289, 0, 0,
1.06079825768261876375865275289, 1.92383174007866700534331407499, 2.28031633126246932878838666430, 2.63039928770751640563098317364, 3.36147409196601931801549610534, 3.89103525500661360235787154837, 4.24134240221525463644226823929, 4.69637272779172553927353758102, 4.98769286084783640906621018650, 5.14549908841918639026426857201, 5.79599136724861448291668108596, 5.87937415617397616000272474492, 6.40676274655494083377340544487, 6.97289060689875183716619875266, 7.00415047295711269708368840159, 7.56581651951199556792667004157, 8.023997210289222093489070038720, 8.043672229136358257505826123348
