| L(s) = 1 | + 3·2-s + 3·3-s + 3·4-s + 9·6-s + 9·12-s − 3·16-s − 6·17-s + 6·23-s − 10·27-s − 15·29-s − 9·31-s − 6·32-s − 18·34-s − 12·41-s + 18·46-s + 6·47-s − 9·48-s − 18·49-s − 18·51-s + 6·53-s − 30·54-s − 45·58-s − 21·59-s + 9·61-s − 27·62-s − 8·64-s − 18·67-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3/2·4-s + 3.67·6-s + 2.59·12-s − 3/4·16-s − 1.45·17-s + 1.25·23-s − 1.92·27-s − 2.78·29-s − 1.61·31-s − 1.06·32-s − 3.08·34-s − 1.87·41-s + 2.65·46-s + 0.875·47-s − 1.29·48-s − 2.57·49-s − 2.52·51-s + 0.824·53-s − 4.08·54-s − 5.90·58-s − 2.73·59-s + 1.15·61-s − 3.42·62-s − 64-s − 2.19·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 17 T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 18 T^{2} - T^{3} + 18 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 24 T^{2} - 9 T^{3} + 24 p T^{4} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 18 T^{2} - 37 T^{3} + 18 p T^{4} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 252 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 15 T + 159 T^{2} + 981 T^{3} + 159 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 9 T + 99 T^{2} + 505 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 90 T^{2} + 17 T^{3} + 90 p T^{4} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 873 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 72 T^{2} - 163 T^{3} + 72 p T^{4} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 567 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 585 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 21 T + 312 T^{2} + 2745 T^{3} + 312 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 9 T + 162 T^{2} - 917 T^{3} + 162 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 18 T + 225 T^{2} + 1988 T^{3} + 225 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 30 T + 501 T^{2} + 5148 T^{3} + 501 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 171 T^{2} - 64 T^{3} + 171 p T^{4} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 9 T + 135 T^{2} + 613 T^{3} + 135 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 60 T^{2} - 459 T^{3} + 60 p T^{4} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 15 T + 321 T^{2} + 2727 T^{3} + 321 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 15 T + 330 T^{2} + 2783 T^{3} + 330 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.20469350116626704163153389222, −6.90921109641727918950653601022, −6.66548801468379738380472100628, −6.48554744128790200367634023619, −6.04616315411545173537236151768, −5.89166704471333517417298600072, −5.77028895146736461244207469726, −5.37484323440490948371739702812, −5.21135526925720556668564727879, −5.20093156421097167411378521570, −4.65813670725426997870416953438, −4.55643986513866117333391146436, −4.40559864777419416153700585770, −3.95455981197836486172419046472, −3.81639961389238012096149850140, −3.75979334523763783297008179712, −3.15356517680023558035527260665, −3.12823769975670429159702933549, −3.09747332313288361023811027438, −2.64109464290453711025156452484, −2.55606242364808288514830478769, −1.84616127988655110124327152962, −1.77718920947393819468622738212, −1.73436463309260037897272512243, −1.17183708093239053539210696974, 0, 0, 0,
1.17183708093239053539210696974, 1.73436463309260037897272512243, 1.77718920947393819468622738212, 1.84616127988655110124327152962, 2.55606242364808288514830478769, 2.64109464290453711025156452484, 3.09747332313288361023811027438, 3.12823769975670429159702933549, 3.15356517680023558035527260665, 3.75979334523763783297008179712, 3.81639961389238012096149850140, 3.95455981197836486172419046472, 4.40559864777419416153700585770, 4.55643986513866117333391146436, 4.65813670725426997870416953438, 5.20093156421097167411378521570, 5.21135526925720556668564727879, 5.37484323440490948371739702812, 5.77028895146736461244207469726, 5.89166704471333517417298600072, 6.04616315411545173537236151768, 6.48554744128790200367634023619, 6.66548801468379738380472100628, 6.90921109641727918950653601022, 7.20469350116626704163153389222
