| L(s) = 1 | − 2·3-s + 5-s + 7·7-s − 9-s − 3·13-s − 2·15-s − 4·17-s − 8·19-s − 14·21-s + 9·23-s − 9·25-s + 7·27-s + 9·29-s + 17·31-s + 7·35-s + 3·37-s + 6·39-s − 16·41-s + 4·43-s − 45-s + 11·47-s + 18·49-s + 8·51-s + 3·53-s + 16·57-s + 2·59-s − 15·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 2.64·7-s − 1/3·9-s − 0.832·13-s − 0.516·15-s − 0.970·17-s − 1.83·19-s − 3.05·21-s + 1.87·23-s − 9/5·25-s + 1.34·27-s + 1.67·29-s + 3.05·31-s + 1.18·35-s + 0.493·37-s + 0.960·39-s − 2.49·41-s + 0.609·43-s − 0.149·45-s + 1.60·47-s + 18/7·49-s + 1.12·51-s + 0.412·53-s + 2.11·57-s + 0.260·59-s − 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.303888963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303888963\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 12 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - p T + 31 T^{2} - 94 T^{3} + 31 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T^{2} - 27 T^{3} - 3 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} + 16 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 31 T^{2} + 120 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 240 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 4 p T^{2} - 428 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 9 T + 110 T^{2} - 536 T^{3} + 110 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 17 T + 184 T^{2} - 1202 T^{3} + 184 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 16 T + 193 T^{2} + 1359 T^{3} + 193 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 9 T^{2} - 112 T^{3} + 9 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 11 T + 131 T^{2} - 1030 T^{3} + 131 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 59 T^{2} - 610 T^{3} + 59 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 53 T^{2} - 220 T^{3} + 53 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 15 T + 212 T^{2} + 1778 T^{3} + 212 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 5 T + 22 T^{2} + 274 T^{3} + 22 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 5 T + 189 T^{2} + 714 T^{3} + 189 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 195 T^{2} + 839 T^{3} + 195 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + T + 218 T^{2} + 126 T^{3} + 218 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 9 T + 173 T^{2} - 1606 T^{3} + 173 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 2784 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 47 T^{2} + 256 T^{3} + 47 p T^{4} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.100305383017282649306717166792, −7.87713980674410054310136891168, −7.37996528113351292139638697311, −7.23117796287311182617567450480, −6.69271501130898435024393329748, −6.67936014207665369751703314947, −6.41056871307618984602622840584, −6.07911294333867206062948173002, −5.73785630886588929384299241940, −5.73547294605903096748859079504, −5.00354101054876286324845177828, −4.94714307768747600552402833158, −4.94692358345736362828616218617, −4.58682374676429556232419766593, −4.27231575117588978167928527891, −4.22507816211424965484808159191, −3.59980817517105006491822333411, −2.92659046104828410001239624764, −2.84472565734398507225682938036, −2.38593516700375032496695090477, −2.18638871977855052714739542336, −1.68710403698863622544783417789, −1.43906942990589203323675081401, −0.845578287138297305164697677004, −0.42366469626933982188294420481,
0.42366469626933982188294420481, 0.845578287138297305164697677004, 1.43906942990589203323675081401, 1.68710403698863622544783417789, 2.18638871977855052714739542336, 2.38593516700375032496695090477, 2.84472565734398507225682938036, 2.92659046104828410001239624764, 3.59980817517105006491822333411, 4.22507816211424965484808159191, 4.27231575117588978167928527891, 4.58682374676429556232419766593, 4.94692358345736362828616218617, 4.94714307768747600552402833158, 5.00354101054876286324845177828, 5.73547294605903096748859079504, 5.73785630886588929384299241940, 6.07911294333867206062948173002, 6.41056871307618984602622840584, 6.67936014207665369751703314947, 6.69271501130898435024393329748, 7.23117796287311182617567450480, 7.37996528113351292139638697311, 7.87713980674410054310136891168, 8.100305383017282649306717166792
