| L(s) = 1 | + 2·3-s + 2·7-s − 2·9-s − 4·11-s − 8·13-s − 4·17-s − 6·19-s + 4·21-s − 3·23-s − 7·27-s − 8·29-s − 20·31-s − 8·33-s − 2·37-s − 16·39-s + 4·41-s + 4·43-s + 6·47-s − 49-s − 8·51-s − 8·53-s − 12·57-s − 9·59-s − 22·61-s − 4·63-s − 4·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s − 2/3·9-s − 1.20·11-s − 2.21·13-s − 0.970·17-s − 1.37·19-s + 0.872·21-s − 0.625·23-s − 1.34·27-s − 1.48·29-s − 3.59·31-s − 1.39·33-s − 0.328·37-s − 2.56·39-s + 0.624·41-s + 0.609·43-s + 0.875·47-s − 1/7·49-s − 1.12·51-s − 1.09·53-s − 1.58·57-s − 1.17·59-s − 2.81·61-s − 0.503·63-s − 0.488·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - p^{2} T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 20 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 21 T^{2} + 64 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 48 T^{2} + 181 T^{3} + 48 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 31 T^{2} + 64 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 172 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 8 T + 44 T^{2} + 207 T^{3} + 44 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 20 T + 206 T^{2} + 1367 T^{3} + 206 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 95 T^{2} + 140 T^{3} + 95 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 16 T^{2} - 7 T^{3} + 16 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 85 T^{2} - 176 T^{3} + 85 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 72 T^{2} - 437 T^{3} + 72 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 75 T^{2} + 920 T^{3} + 75 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 9 T + 96 T^{2} + 341 T^{3} + 96 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 22 T + 295 T^{2} + 2612 T^{3} + 295 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T - 51 T^{2} - 640 T^{3} - 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 30 T + 504 T^{2} + 5171 T^{3} + 504 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 284 T^{2} - 2413 T^{3} + 284 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 20 T + 321 T^{2} + 3232 T^{3} + 321 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 245 T^{2} + 1320 T^{3} + 245 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 171 T^{2} - 1284 T^{3} + 171 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 263 T^{2} - 1544 T^{3} + 263 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.547170024594674299516541945942, −7.83711637865878757427796665337, −7.78453941962996814225825887309, −7.75951433706526903475884415102, −7.38501102735218851055155146522, −7.35976052358875308384113183537, −7.13006761147834870615376146754, −6.40797774396680587911651123073, −6.23711716941591318654441192325, −6.12000744902824062889527034354, −5.50444494512280510922312128673, −5.47359337619895615797619993667, −5.29893687983539820372578129304, −4.81772607791398949355989310014, −4.57472758781198945227058904944, −4.27229827307947312684433557941, −4.14884968201246521372164031333, −3.50558641034007953980630349557, −3.39940030153480004822834839744, −2.85350853558613684656045820417, −2.70178882735733829876154172392, −2.39557735051727559153836807214, −2.08054658192431456112217589375, −1.72309072459831302226912464321, −1.59928061157995782030515106388, 0, 0, 0,
1.59928061157995782030515106388, 1.72309072459831302226912464321, 2.08054658192431456112217589375, 2.39557735051727559153836807214, 2.70178882735733829876154172392, 2.85350853558613684656045820417, 3.39940030153480004822834839744, 3.50558641034007953980630349557, 4.14884968201246521372164031333, 4.27229827307947312684433557941, 4.57472758781198945227058904944, 4.81772607791398949355989310014, 5.29893687983539820372578129304, 5.47359337619895615797619993667, 5.50444494512280510922312128673, 6.12000744902824062889527034354, 6.23711716941591318654441192325, 6.40797774396680587911651123073, 7.13006761147834870615376146754, 7.35976052358875308384113183537, 7.38501102735218851055155146522, 7.75951433706526903475884415102, 7.78453941962996814225825887309, 7.83711637865878757427796665337, 8.547170024594674299516541945942
