| L(s) = 1 | − 2-s − 2·5-s + 6·7-s − 2·8-s − 5·9-s + 2·10-s + 2·11-s − 2·13-s − 6·14-s + 3·16-s − 6·17-s + 5·18-s + 4·19-s − 2·22-s + 4·23-s − 7·25-s + 2·26-s − 2·27-s − 6·29-s + 16·31-s − 3·32-s + 6·34-s − 12·35-s − 6·37-s − 4·38-s + 4·40-s + 3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s + 2.26·7-s − 0.707·8-s − 5/3·9-s + 0.632·10-s + 0.603·11-s − 0.554·13-s − 1.60·14-s + 3/4·16-s − 1.45·17-s + 1.17·18-s + 0.917·19-s − 0.426·22-s + 0.834·23-s − 7/5·25-s + 0.392·26-s − 0.384·27-s − 1.11·29-s + 2.87·31-s − 0.530·32-s + 1.02·34-s − 2.02·35-s − 0.986·37-s − 0.648·38-s + 0.632·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68921 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68921 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3181260746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3181260746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + T^{2} + 3 T^{3} + p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 11 T^{2} + 16 T^{3} + 11 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 6 T + 29 T^{2} - 86 T^{3} + 29 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} + 6 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 41 T^{2} - 162 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 37 T^{2} - 216 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 83 T^{2} + 308 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 16 T + 157 T^{2} - 1024 T^{3} + 157 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 336 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 121 T^{2} + 328 T^{3} + 121 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 502 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 155 T^{2} - 628 T^{3} + 155 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 161 T^{2} + 784 T^{3} + 161 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 131 T^{2} - 60 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 181 T^{2} + 218 T^{3} + 181 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 20 T + 297 T^{2} - 2706 T^{3} + 297 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2 T + 39 T^{2} + 536 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 32 T + 565 T^{2} - 6146 T^{3} + 565 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 185 T^{2} - 128 T^{3} + 185 p T^{4} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 119 T^{2} + 148 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 239 T^{2} - 916 T^{3} + 239 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.91336200523809483351601555541, −14.22253080403673396082387708383, −13.92225426122259878414512191078, −13.66946044047958969524487815118, −13.13170536541987678948922981983, −12.04937815819870602134998117258, −12.02697670630786306497922221880, −11.86952377845177227169703056383, −11.26569872319727172613840710601, −10.99349616235852862205907573115, −10.94269555113269070169198308114, −9.719343105570443308267388550378, −9.644025101467508164016452203628, −8.903833403877222114675285043380, −8.583196060793129675853966373770, −8.308711178390771027808150974372, −7.71870526716192346254788143179, −7.63196582345436066291380439632, −6.61041296440600507604659986604, −6.15594604372841359728162390431, −5.19303412010009912893951517847, −5.05968586475281254475232400968, −4.19259081075605077803043927645, −3.33309696303425523695901098260, −2.20228565748956291852367527713,
2.20228565748956291852367527713, 3.33309696303425523695901098260, 4.19259081075605077803043927645, 5.05968586475281254475232400968, 5.19303412010009912893951517847, 6.15594604372841359728162390431, 6.61041296440600507604659986604, 7.63196582345436066291380439632, 7.71870526716192346254788143179, 8.308711178390771027808150974372, 8.583196060793129675853966373770, 8.903833403877222114675285043380, 9.644025101467508164016452203628, 9.719343105570443308267388550378, 10.94269555113269070169198308114, 10.99349616235852862205907573115, 11.26569872319727172613840710601, 11.86952377845177227169703056383, 12.02697670630786306497922221880, 12.04937815819870602134998117258, 13.13170536541987678948922981983, 13.66946044047958969524487815118, 13.92225426122259878414512191078, 14.22253080403673396082387708383, 14.91336200523809483351601555541
