| L(s) = 1 | − 2·4-s + 2·7-s − 3·9-s − 12·13-s + 3·16-s − 8·19-s + 25-s − 4·28-s − 22·31-s + 6·36-s + 24·37-s + 12·43-s + 15·49-s + 24·52-s − 6·63-s − 4·64-s + 4·67-s + 24·73-s + 16·76-s − 36·79-s − 24·91-s − 4·97-s − 2·100-s − 14·103-s − 16·109-s + 6·112-s + 36·117-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s − 9-s − 3.32·13-s + 3/4·16-s − 1.83·19-s + 1/5·25-s − 0.755·28-s − 3.95·31-s + 36-s + 3.94·37-s + 1.82·43-s + 15/7·49-s + 3.32·52-s − 0.755·63-s − 1/2·64-s + 0.488·67-s + 2.80·73-s + 1.83·76-s − 4.05·79-s − 2.51·91-s − 0.406·97-s − 1/5·100-s − 1.37·103-s − 1.53·109-s + 0.566·112-s + 3.32·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3613686953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3613686953\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 5 T^{2} - 96 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 46 T^{2} + 435 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 41 T^{2} - 1128 T^{4} + 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 107 T^{2} + 7968 T^{4} - 107 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 2 T^{2} - 5037 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 18 T + 187 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 139 T^{2} + 12432 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
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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
−7.34634454357223761229482112669, −7.11886905820432567935058806394, −6.72153880524251968358724154211, −6.68616218034412165322903133104, −6.34574342492526309749271101235, −5.73745379082886703804657323519, −5.71235128293159518809124111718, −5.57038186957210885266663455092, −5.49180064158375595579167208877, −5.31518518774862399134446897733, −4.63676246447343807603903998229, −4.63515798158073784367472199683, −4.38238202693300249271918285502, −4.30560508193028983994449857092, −4.08815879434775405704147675419, −3.64273198799905050629209685702, −3.35137640370174772292925367158, −2.85121877304039079930924661469, −2.71598343930472018465862540324, −2.28910609151259463945568030526, −2.21299922716708077311597820150, −2.02667978665344658950777078887, −1.30504364245393766772425949845, −0.71439241652872735583149676646, −0.19614357898008788258020947211,
0.19614357898008788258020947211, 0.71439241652872735583149676646, 1.30504364245393766772425949845, 2.02667978665344658950777078887, 2.21299922716708077311597820150, 2.28910609151259463945568030526, 2.71598343930472018465862540324, 2.85121877304039079930924661469, 3.35137640370174772292925367158, 3.64273198799905050629209685702, 4.08815879434775405704147675419, 4.30560508193028983994449857092, 4.38238202693300249271918285502, 4.63515798158073784367472199683, 4.63676246447343807603903998229, 5.31518518774862399134446897733, 5.49180064158375595579167208877, 5.57038186957210885266663455092, 5.71235128293159518809124111718, 5.73745379082886703804657323519, 6.34574342492526309749271101235, 6.68616218034412165322903133104, 6.72153880524251968358724154211, 7.11886905820432567935058806394, 7.34634454357223761229482112669
