| L(s) = 1 | − 3·3-s − 4·5-s − 2·7-s + 6·9-s − 4·11-s − 8·13-s + 12·15-s + 2·17-s + 2·19-s + 6·21-s + 10·23-s + 3·25-s − 10·27-s + 6·29-s + 2·31-s + 12·33-s + 8·35-s − 20·37-s + 24·39-s + 3·41-s + 10·43-s − 24·45-s − 4·47-s − 3·49-s − 6·51-s − 14·53-s + 16·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s − 0.755·7-s + 2·9-s − 1.20·11-s − 2.21·13-s + 3.09·15-s + 0.485·17-s + 0.458·19-s + 1.30·21-s + 2.08·23-s + 3/5·25-s − 1.92·27-s + 1.11·29-s + 0.359·31-s + 2.08·33-s + 1.35·35-s − 3.28·37-s + 3.84·39-s + 0.468·41-s + 1.52·43-s − 3.57·45-s − 0.583·47-s − 3/7·49-s − 0.840·51-s − 1.92·53-s + 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 4 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 34 T^{2} + 84 T^{3} + 34 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 204 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 28 T^{2} - 6 T^{3} + 28 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 51 T^{2} - 68 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 95 T^{2} - 476 T^{3} + 95 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 262 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 2 T^{2} + 132 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 20 T + 228 T^{2} + 1646 T^{3} + 228 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 10 T^{2} + 296 T^{3} + 10 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 106 T^{2} + 384 T^{3} + 106 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 3 p T^{2} + 1452 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 137 T^{2} + 976 T^{3} + 137 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 188 T^{2} - 930 T^{3} + 188 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 632 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 32 T + 550 T^{2} - 5712 T^{3} + 550 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T + 120 T^{2} - 130 T^{3} + 120 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 20 T + 305 T^{2} - 3192 T^{3} + 305 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 259 T^{2} + 2028 T^{3} + 259 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 12 T + 305 T^{2} + 2180 T^{3} + 305 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.28368948400010981154111631012, −7.05394680641895001785814460489, −6.79086852774315360181609811731, −6.60608821085780313863149726946, −6.47312813402108662374415802913, −6.21606639353798919663010669346, −5.90907384062927255005408350180, −5.40033389549063158264225937899, −5.20607883082215704327256810529, −5.18325054211735931776231806682, −4.93650208931679376957773039677, −4.88351052374331310786331548937, −4.71198328651986241952343102811, −4.09306470480226254987970900870, −3.90422316219219417358975533847, −3.79504511757109823846142730318, −3.37847547005482715475919982652, −3.09177715894576785685919277202, −3.07178560002439088197466077009, −2.42122133991537993222977854985, −2.28483668483585408518394171268, −2.12766237538172263938642158532, −1.22812758327462589453907572685, −1.14092440549739283284241338430, −0.75139157471455870132628303655, 0, 0, 0,
0.75139157471455870132628303655, 1.14092440549739283284241338430, 1.22812758327462589453907572685, 2.12766237538172263938642158532, 2.28483668483585408518394171268, 2.42122133991537993222977854985, 3.07178560002439088197466077009, 3.09177715894576785685919277202, 3.37847547005482715475919982652, 3.79504511757109823846142730318, 3.90422316219219417358975533847, 4.09306470480226254987970900870, 4.71198328651986241952343102811, 4.88351052374331310786331548937, 4.93650208931679376957773039677, 5.18325054211735931776231806682, 5.20607883082215704327256810529, 5.40033389549063158264225937899, 5.90907384062927255005408350180, 6.21606639353798919663010669346, 6.47312813402108662374415802913, 6.60608821085780313863149726946, 6.79086852774315360181609811731, 7.05394680641895001785814460489, 7.28368948400010981154111631012
