| L(s) = 1 | − 2·7-s + 16-s − 2·19-s − 4·31-s + 2·37-s + 2·49-s − 2·67-s + 4·73-s + 2·97-s + 4·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2·7-s + 16-s − 2·19-s − 4·31-s + 2·37-s + 2·49-s − 2·67-s + 4·73-s + 2·97-s + 4·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4529463966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4529463966\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12164406315531926627217705002, −6.45659486016403027366744578034, −6.44100172084808297158730745855, −6.30712022747573352037572162635, −6.25147086243226609438728191415, −6.01438443642342500997404719769, −5.75846590574526933059564277531, −5.45057657107226872241429111748, −5.25475323271733295219949286420, −5.02411477375288570666007554848, −4.81268408536586929100584542989, −4.48212007270362702465734413992, −4.24616194660529776666254298672, −3.79938743537591718310803492437, −3.73748892601373558384545000089, −3.54521202356348720887550044278, −3.51792459661940937843754039878, −3.15171253069263442518615307088, −2.57816931547142075901048643722, −2.53367718792264570517182160135, −2.26939017078099746105840395363, −1.98472107510920916283955765044, −1.51196050793068896330305366652, −1.15383377752205313082012180048, −0.41723163149666320861635498051,
0.41723163149666320861635498051, 1.15383377752205313082012180048, 1.51196050793068896330305366652, 1.98472107510920916283955765044, 2.26939017078099746105840395363, 2.53367718792264570517182160135, 2.57816931547142075901048643722, 3.15171253069263442518615307088, 3.51792459661940937843754039878, 3.54521202356348720887550044278, 3.73748892601373558384545000089, 3.79938743537591718310803492437, 4.24616194660529776666254298672, 4.48212007270362702465734413992, 4.81268408536586929100584542989, 5.02411477375288570666007554848, 5.25475323271733295219949286420, 5.45057657107226872241429111748, 5.75846590574526933059564277531, 6.01438443642342500997404719769, 6.25147086243226609438728191415, 6.30712022747573352037572162635, 6.44100172084808297158730745855, 6.45659486016403027366744578034, 7.12164406315531926627217705002
