| L(s) = 1 | − 2·4-s + 2·7-s + 4·16-s + 6·19-s − 4·28-s + 6·31-s − 2·37-s + 4·43-s − 11·49-s − 12·61-s − 16·64-s + 22·67-s − 6·73-s − 12·76-s − 10·79-s + 30·103-s + 2·109-s + 8·112-s − 20·121-s − 12·124-s + 127-s + 131-s + 12·133-s + 137-s + 139-s + 4·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s + 16-s + 1.37·19-s − 0.755·28-s + 1.07·31-s − 0.328·37-s + 0.609·43-s − 1.57·49-s − 1.53·61-s − 2·64-s + 2.68·67-s − 0.702·73-s − 1.37·76-s − 1.12·79-s + 2.95·103-s + 0.191·109-s + 0.755·112-s − 1.81·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 1.04·133-s + 0.0854·137-s + 0.0848·139-s + 0.328·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.597261584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.597261584\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 + 20 T^{2} + 279 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 56 T^{2} + 927 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 98 T^{2} + 6795 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 94 T^{2} + 5355 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 154 T^{2} + 15795 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.82106290275388085134360760601, −6.35376972859363874496400785043, −6.21516372780240403803184983138, −6.18722293082823443726536666062, −5.78215465332680794027871270825, −5.57229399138998772021144677362, −5.43534464080181039957380194063, −5.06127261786452702038868299782, −4.94082391546738616133489776249, −4.80890258032329613704687775350, −4.59180089087665299869702336813, −4.30855247604749914596445666907, −4.14258149937351857663980238326, −3.75118067228510691583069313338, −3.46584389713184676935119273367, −3.30459046626302054501277281538, −3.29581494214728492448581621492, −2.56481308501511236256083756647, −2.54516796254332030829960636223, −2.37687159569534293598171328943, −1.65619169685046620790635569729, −1.41187875042519474911564691397, −1.31014691201399497390169003224, −0.835641293230984037778465550312, −0.27587541648286738947509942211,
0.27587541648286738947509942211, 0.835641293230984037778465550312, 1.31014691201399497390169003224, 1.41187875042519474911564691397, 1.65619169685046620790635569729, 2.37687159569534293598171328943, 2.54516796254332030829960636223, 2.56481308501511236256083756647, 3.29581494214728492448581621492, 3.30459046626302054501277281538, 3.46584389713184676935119273367, 3.75118067228510691583069313338, 4.14258149937351857663980238326, 4.30855247604749914596445666907, 4.59180089087665299869702336813, 4.80890258032329613704687775350, 4.94082391546738616133489776249, 5.06127261786452702038868299782, 5.43534464080181039957380194063, 5.57229399138998772021144677362, 5.78215465332680794027871270825, 6.18722293082823443726536666062, 6.21516372780240403803184983138, 6.35376972859363874496400785043, 6.82106290275388085134360760601
