| L(s) = 1 | − 3-s − 7-s − 4·9-s + 11·13-s + 3·17-s − 3·19-s + 21-s − 9·23-s + 5·27-s − 7·29-s − 6·31-s + 20·37-s − 11·39-s − 22·41-s − 10·43-s − 4·49-s − 3·51-s + 7·53-s + 3·57-s − 11·59-s − 16·61-s + 4·63-s − 67-s + 9·69-s + 5·73-s − 26·79-s + 4·81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 4/3·9-s + 3.05·13-s + 0.727·17-s − 0.688·19-s + 0.218·21-s − 1.87·23-s + 0.962·27-s − 1.29·29-s − 1.07·31-s + 3.28·37-s − 1.76·39-s − 3.43·41-s − 1.52·43-s − 4/7·49-s − 0.420·51-s + 0.961·53-s + 0.397·57-s − 1.43·59-s − 2.04·61-s + 0.503·63-s − 0.122·67-s + 1.08·69-s + 0.585·73-s − 2.92·79-s + 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 4 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 11 T + 55 T^{2} - 200 T^{3} + 55 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 47 T^{2} - 98 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 89 T^{2} + 422 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 43 T^{2} + 114 T^{3} + 43 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 340 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 20 T + 237 T^{2} - 1724 T^{3} + 237 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $D_{6}$ | \( 1 + 22 T + 223 T^{2} + 1572 T^{3} + 223 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 97 T^{2} + 508 T^{3} + 97 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 29 T^{2} - 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 7 T - 9 T^{2} + 600 T^{3} - 9 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 11 T + 37 T^{2} - 246 T^{3} + 37 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 16 T + 207 T^{2} + 1600 T^{3} + 207 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 101 T^{2} + 396 T^{3} + 101 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 - 5 T + 47 T^{2} + 498 T^{3} + 47 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 26 T + 445 T^{2} + 4604 T^{3} + 445 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 297 T^{2} + 2340 T^{3} + 297 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} - 188 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 233 T^{2} + 1260 T^{3} + 233 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.42877596048540592806071141937, −6.83772685236564348512512202052, −6.70151378569186683147444614844, −6.67896131610645873239987989115, −6.25035405877465807218725708130, −6.00885357503012973861436867063, −5.90514988057904074176330268870, −5.71482561945000762472362277464, −5.61467721026013841803444097749, −5.51959799885063156798328309927, −4.84953238165544099428324539667, −4.73871983283688822220429205816, −4.40713077012735598935321776084, −4.14191629345835874962304382767, −3.93619116967660609852633644907, −3.61904644300565840880373006208, −3.25796457941323063961173566451, −3.18368769000238892236909614672, −3.17816962928387280139731925173, −2.56127106596545953737272097955, −2.08220868731188979112641009485, −2.01764889940170312058276348551, −1.40216878621399549303345019019, −1.35002714741097480850818413161, −1.06023726517714470950466486405, 0, 0, 0,
1.06023726517714470950466486405, 1.35002714741097480850818413161, 1.40216878621399549303345019019, 2.01764889940170312058276348551, 2.08220868731188979112641009485, 2.56127106596545953737272097955, 3.17816962928387280139731925173, 3.18368769000238892236909614672, 3.25796457941323063961173566451, 3.61904644300565840880373006208, 3.93619116967660609852633644907, 4.14191629345835874962304382767, 4.40713077012735598935321776084, 4.73871983283688822220429205816, 4.84953238165544099428324539667, 5.51959799885063156798328309927, 5.61467721026013841803444097749, 5.71482561945000762472362277464, 5.90514988057904074176330268870, 6.00885357503012973861436867063, 6.25035405877465807218725708130, 6.67896131610645873239987989115, 6.70151378569186683147444614844, 6.83772685236564348512512202052, 7.42877596048540592806071141937
