| L(s) = 1 | − 3-s − 5-s + 4·7-s + 2·9-s + 5·11-s − 5·13-s + 15-s + 2·17-s + 3·19-s − 4·21-s − 5·23-s − 4·25-s − 27-s + 9·29-s − 5·33-s − 4·35-s + 6·37-s + 5·39-s + 8·41-s − 17·43-s − 2·45-s − 47-s − 2·51-s − 53-s − 5·55-s − 3·57-s + 23·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 2/3·9-s + 1.50·11-s − 1.38·13-s + 0.258·15-s + 0.485·17-s + 0.688·19-s − 0.872·21-s − 1.04·23-s − 4/5·25-s − 0.192·27-s + 1.67·29-s − 0.870·33-s − 0.676·35-s + 0.986·37-s + 0.800·39-s + 1.24·41-s − 2.59·43-s − 0.298·45-s − 0.145·47-s − 0.280·51-s − 0.137·53-s − 0.674·55-s − 0.397·57-s + 2.99·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.639111318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.639111318\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $A_4\times C_2$ | \( 1 + T - T^{2} - 2 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 4 T + 16 T^{2} - 40 T^{3} + 16 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 31 T^{2} - 102 T^{3} + 31 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 5 T + 37 T^{2} + 122 T^{3} + 37 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 2 T + 42 T^{2} - 66 T^{3} + 42 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 5 T + 5 T^{2} - 26 T^{3} + 5 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 9 T + 83 T^{2} - 518 T^{3} + 83 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 41 | $A_4\times C_2$ | \( 1 - 8 T + 103 T^{2} - 528 T^{3} + 103 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 17 T + 153 T^{2} + 1094 T^{3} + 153 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + T + 69 T^{2} - 162 T^{3} + 69 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + T + 25 T^{2} - 150 T^{3} + 25 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 23 T + 343 T^{2} - 3090 T^{3} + 343 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 3 T + 155 T^{2} + 274 T^{3} + 155 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 15 T + 245 T^{2} + 2042 T^{3} + 245 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 12 T + 137 T^{2} + 776 T^{3} + 137 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 4 T + 152 T^{2} - 258 T^{3} + 152 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 26 T + 421 T^{2} - 4364 T^{3} + 421 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 260 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_6$ | \( 1 - 18 T + 251 T^{2} - 2180 T^{3} + 251 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 8 T + 271 T^{2} + 1424 T^{3} + 271 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.724692215891344721254441414291, −8.191261480585018046320310898374, −8.045492338539061940288885382852, −7.963477130762389877664919544672, −7.49892888563301805436477418141, −7.38668316735586331506639782965, −6.91011933181459577598289719745, −6.77457981668460925208149187536, −6.36152054071524836373729524539, −6.25235473640743344254037613069, −5.67715545051411091771081392462, −5.57368513645880235519978933978, −5.05491666030481672902755041173, −4.85154303957576770722313795609, −4.51785979064268742069346790353, −4.49972233458906706416724590528, −3.96724631199645914019872578227, −3.56861002067600906914453846833, −3.45316413083897154696901349520, −2.71807828623708951045835421700, −2.41582926068248001397405527341, −1.86873765549236741581451181933, −1.56230628834506061117365327499, −1.09485038587736005961420078244, −0.54084436357386275603258832822,
0.54084436357386275603258832822, 1.09485038587736005961420078244, 1.56230628834506061117365327499, 1.86873765549236741581451181933, 2.41582926068248001397405527341, 2.71807828623708951045835421700, 3.45316413083897154696901349520, 3.56861002067600906914453846833, 3.96724631199645914019872578227, 4.49972233458906706416724590528, 4.51785979064268742069346790353, 4.85154303957576770722313795609, 5.05491666030481672902755041173, 5.57368513645880235519978933978, 5.67715545051411091771081392462, 6.25235473640743344254037613069, 6.36152054071524836373729524539, 6.77457981668460925208149187536, 6.91011933181459577598289719745, 7.38668316735586331506639782965, 7.49892888563301805436477418141, 7.963477130762389877664919544672, 8.045492338539061940288885382852, 8.191261480585018046320310898374, 8.724692215891344721254441414291
