| L(s) = 1 | − 2·3-s + 3·5-s − 4·7-s − 9-s − 2·11-s − 4·13-s − 6·15-s + 6·17-s + 3·19-s + 8·21-s − 8·23-s + 6·25-s + 2·27-s − 2·29-s − 2·31-s + 4·33-s − 12·35-s − 2·37-s + 8·39-s + 16·41-s − 8·43-s − 3·45-s − 4·47-s − 49-s − 12·51-s − 10·53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s − 1.51·7-s − 1/3·9-s − 0.603·11-s − 1.10·13-s − 1.54·15-s + 1.45·17-s + 0.688·19-s + 1.74·21-s − 1.66·23-s + 6/5·25-s + 0.384·27-s − 0.371·29-s − 0.359·31-s + 0.696·33-s − 2.02·35-s − 0.328·37-s + 1.28·39-s + 2.49·41-s − 1.21·43-s − 0.447·45-s − 0.583·47-s − 1/7·49-s − 1.68·51-s − 1.37·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 52 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 25 T^{2} + 32 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 33 T^{2} + 86 T^{3} + 33 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 81 T^{2} + 356 T^{3} + 81 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 67 T^{2} + 92 T^{3} + 67 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 73 T^{2} + 100 T^{3} + 73 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 101 T^{2} + 146 T^{3} + 101 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 16 T + 187 T^{2} - 1360 T^{3} + 187 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 113 T^{2} + 524 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 137 T^{2} + 372 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 181 T^{2} + 1054 T^{3} + 181 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 260 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 59 T^{2} + 8 p T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 69 T^{2} - 238 T^{3} + 69 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 26 T + 417 T^{2} + 4148 T^{3} + 417 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 199 T^{2} - 268 T^{3} + 199 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 273 T^{2} + 1996 T^{3} + 273 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 26 T + 471 T^{2} - 5084 T^{3} + 471 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 381 T^{2} - 3574 T^{3} + 381 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.50628090654188994787182695330, −7.43104157976422137842862458159, −6.83735940686721603615685659732, −6.77032996011102969807019121939, −6.23644400771493316606274566123, −6.14278695914384389089726935806, −6.12618270916358074101409366346, −5.82626470158477895417442188207, −5.70837883036647392327581983014, −5.56450529161761230498693839451, −4.99139999841812444166902904393, −4.90139360595583994505550941392, −4.80780749561143823223321246677, −4.56403000563324265585509032237, −3.91260233418145068974188021734, −3.73388024444353338280112966497, −3.33259525588562444012583992444, −3.21618516120149464202367203932, −3.07708777418381273014530681750, −2.47586530452988934410423216348, −2.35221382086916147270827997690, −2.16074709398352134144706411677, −1.64744287366201213112185112984, −1.18249750225369321640110377319, −1.02743142026362663927319399022, 0, 0, 0,
1.02743142026362663927319399022, 1.18249750225369321640110377319, 1.64744287366201213112185112984, 2.16074709398352134144706411677, 2.35221382086916147270827997690, 2.47586530452988934410423216348, 3.07708777418381273014530681750, 3.21618516120149464202367203932, 3.33259525588562444012583992444, 3.73388024444353338280112966497, 3.91260233418145068974188021734, 4.56403000563324265585509032237, 4.80780749561143823223321246677, 4.90139360595583994505550941392, 4.99139999841812444166902904393, 5.56450529161761230498693839451, 5.70837883036647392327581983014, 5.82626470158477895417442188207, 6.12618270916358074101409366346, 6.14278695914384389089726935806, 6.23644400771493316606274566123, 6.77032996011102969807019121939, 6.83735940686721603615685659732, 7.43104157976422137842862458159, 7.50628090654188994787182695330
