| L(s) = 1 | − 2·5-s + 4·11-s + 3·13-s − 6·17-s + 25-s − 6·29-s + 8·31-s + 2·37-s − 10·41-s − 4·43-s − 8·47-s − 5·49-s − 14·53-s − 8·55-s + 20·59-s + 2·61-s − 6·65-s + 16·67-s − 8·71-s + 22·73-s + 36·79-s + 4·83-s + 12·85-s + 6·89-s − 2·97-s − 6·101-s + 28·103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + 0.832·13-s − 1.45·17-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s − 5/7·49-s − 1.92·53-s − 1.07·55-s + 2.60·59-s + 0.256·61-s − 0.744·65-s + 1.95·67-s − 0.949·71-s + 2.57·73-s + 4.05·79-s + 0.439·83-s + 1.30·85-s + 0.635·89-s − 0.203·97-s − 0.597·101-s + 2.75·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.241251136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.241251136\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 41 T^{2} - 16 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 61 T^{2} - 224 T^{3} + 61 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 59 T^{2} - 108 T^{3} + 59 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 143 T^{2} + 812 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 280 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 720 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 171 T^{2} + 1332 T^{3} + 171 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 20 T + 273 T^{2} - 2328 T^{3} + 273 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 16 T + 137 T^{2} - 1104 T^{3} + 137 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 197 T^{2} + 976 T^{3} + 197 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 327 T^{2} - 3220 T^{3} + 327 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 217 T^{2} - 696 T^{3} + 217 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 159 T^{2} - 852 T^{3} + 159 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 239 T^{2} + 348 T^{3} + 239 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.69099032969967378942778418544, −7.19553209127642103216778948927, −6.94813964194907037762720804315, −6.75304339369194624879972398698, −6.53331782152811097324513219951, −6.43164197649586660337400913161, −6.13214351604501098054627347639, −5.98959113837174622272351261304, −5.39699733556648545286361475261, −5.14503806782569889130987537192, −4.89055907670581697658686369348, −4.82265560109441868010551636563, −4.45006438531821323417128805189, −4.02835875087941185941142377188, −3.78716209460794798207333237296, −3.73445369040103819825123941844, −3.29778843805567975282214110505, −3.23102519779990504948855215993, −2.73834803168262196239306174899, −2.01709642598522270603829511084, −1.99931661267714982912968683824, −1.91220421895517642299308082562, −1.14540152506832449270669972735, −0.67172784026083215921599892484, −0.48808188129536633221085891707,
0.48808188129536633221085891707, 0.67172784026083215921599892484, 1.14540152506832449270669972735, 1.91220421895517642299308082562, 1.99931661267714982912968683824, 2.01709642598522270603829511084, 2.73834803168262196239306174899, 3.23102519779990504948855215993, 3.29778843805567975282214110505, 3.73445369040103819825123941844, 3.78716209460794798207333237296, 4.02835875087941185941142377188, 4.45006438531821323417128805189, 4.82265560109441868010551636563, 4.89055907670581697658686369348, 5.14503806782569889130987537192, 5.39699733556648545286361475261, 5.98959113837174622272351261304, 6.13214351604501098054627347639, 6.43164197649586660337400913161, 6.53331782152811097324513219951, 6.75304339369194624879972398698, 6.94813964194907037762720804315, 7.19553209127642103216778948927, 7.69099032969967378942778418544
