| L(s) = 1 | + 2-s − 3-s − 6-s − 8-s + 9-s − 2·11-s − 16-s + 2·17-s + 18-s − 19-s − 2·22-s + 24-s − 2·27-s + 2·33-s + 2·34-s − 38-s + 41-s + 2·43-s + 48-s + 2·49-s − 2·51-s − 2·54-s + 57-s + 59-s + 64-s + 2·66-s − 67-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 6-s − 8-s + 9-s − 2·11-s − 16-s + 2·17-s + 18-s − 19-s − 2·22-s + 24-s − 2·27-s + 2·33-s + 2·34-s − 38-s + 41-s + 2·43-s + 48-s + 2·49-s − 2·51-s − 2·54-s + 57-s + 59-s + 64-s + 2·66-s − 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.010791626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010791626\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.801061667146172223900006302073, −8.455486710762704136513495198753, −7.908190608243251120909372638082, −7.67910070770663830479328246272, −7.46696163752520357293982429056, −6.94944209964752139845463856012, −6.53633365398466010734190753223, −5.91220156149461080410026856878, −5.66467494285632800173711656226, −5.58269055062843969766651745156, −5.32763178529595750794184896519, −4.67232190865074126792521798506, −4.40672017324912073846013203006, −3.91585151042771748337942254484, −3.69640326728159162261851970047, −2.92764953168571422288492782366, −2.61457635491670598605040696977, −2.20386241958171626950647796076, −1.30942933133815676285840403259, −0.54689964200041478865156603202,
0.54689964200041478865156603202, 1.30942933133815676285840403259, 2.20386241958171626950647796076, 2.61457635491670598605040696977, 2.92764953168571422288492782366, 3.69640326728159162261851970047, 3.91585151042771748337942254484, 4.40672017324912073846013203006, 4.67232190865074126792521798506, 5.32763178529595750794184896519, 5.58269055062843969766651745156, 5.66467494285632800173711656226, 5.91220156149461080410026856878, 6.53633365398466010734190753223, 6.94944209964752139845463856012, 7.46696163752520357293982429056, 7.67910070770663830479328246272, 7.908190608243251120909372638082, 8.455486710762704136513495198753, 8.801061667146172223900006302073
