| L(s) = 1 | + 4·9-s − 2·49-s + 10·81-s + 127-s + 131-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 + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | + 4·9-s − 2·49-s + 10·81-s + 127-s + 131-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 + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.881947202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.881947202\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.42095431891894210900164211972, −6.25704832894590949948994758262, −5.95063896988352383909471638250, −5.50601390622913120500388026647, −5.34530639815984334354366400641, −5.15802516343223768273212559177, −4.96326697214951717021633596758, −4.88588452162000120400617885349, −4.67403151682448148823671157555, −4.29941116920293469845474378640, −4.08279796690732106460286932667, −4.07652877546709627477569665315, −4.04727584697386713694811014306, −3.59117934576752853411265059366, −3.41641202176248070030045285624, −3.23574374349493958232952753058, −2.84720679346905027335178056338, −2.58700198493936614235213626002, −2.34670263617491329562798163291, −1.98852164419987799898172843405, −1.73748507093981321275491737274, −1.50872690361613829391944336909, −1.43783836857287208080680197779, −1.04758633771778830653930507040, −0.66440081984730776364344582529,
0.66440081984730776364344582529, 1.04758633771778830653930507040, 1.43783836857287208080680197779, 1.50872690361613829391944336909, 1.73748507093981321275491737274, 1.98852164419987799898172843405, 2.34670263617491329562798163291, 2.58700198493936614235213626002, 2.84720679346905027335178056338, 3.23574374349493958232952753058, 3.41641202176248070030045285624, 3.59117934576752853411265059366, 4.04727584697386713694811014306, 4.07652877546709627477569665315, 4.08279796690732106460286932667, 4.29941116920293469845474378640, 4.67403151682448148823671157555, 4.88588452162000120400617885349, 4.96326697214951717021633596758, 5.15802516343223768273212559177, 5.34530639815984334354366400641, 5.50601390622913120500388026647, 5.95063896988352383909471638250, 6.25704832894590949948994758262, 6.42095431891894210900164211972
