| L(s) = 1 | − 2·2-s − 2·3-s − 4-s − 4·5-s + 4·6-s − 3·7-s + 5·8-s − 4·9-s + 8·10-s − 8·11-s + 2·12-s + 6·14-s + 8·15-s − 16-s − 2·17-s + 8·18-s − 4·19-s + 4·20-s + 6·21-s + 16·22-s − 5·23-s − 10·24-s − 2·25-s + 13·27-s + 3·28-s − 29-s − 16·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s − 1/2·4-s − 1.78·5-s + 1.63·6-s − 1.13·7-s + 1.76·8-s − 4/3·9-s + 2.52·10-s − 2.41·11-s + 0.577·12-s + 1.60·14-s + 2.06·15-s − 1/4·16-s − 0.485·17-s + 1.88·18-s − 0.917·19-s + 0.894·20-s + 1.30·21-s + 3.41·22-s − 1.04·23-s − 2.04·24-s − 2/5·25-s + 2.50·27-s + 0.566·28-s − 0.185·29-s − 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4826809 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4826809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + p T + 5 T^{2} + 7 T^{3} + 5 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 + 2 T + 8 T^{2} + 11 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 4 T + 18 T^{2} + 39 T^{3} + 18 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 17 T^{2} + 29 T^{3} + 17 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 8 T + 52 T^{2} + 189 T^{3} + 52 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 2 T + 36 T^{2} + 81 T^{3} + 36 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 4 T + 46 T^{2} + 151 T^{3} + 46 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 5 T + 68 T^{2} + 217 T^{3} + 68 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + T + 43 T^{2} + 141 T^{3} + 43 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 5 T + 57 T^{2} + 143 T^{3} + 57 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 12 T + 152 T^{2} - 917 T^{3} + 152 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 7 T + 74 T^{2} + 623 T^{3} + 74 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 13 T + 169 T^{2} - 1105 T^{3} + 169 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 18 T + 242 T^{2} + 1859 T^{3} + 242 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - T + 73 T^{2} + 231 T^{3} + 73 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 19 T + 260 T^{2} + 2243 T^{3} + 260 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 4 T + 116 T^{2} - 249 T^{3} + 116 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + T + 129 T^{2} + 175 T^{3} + 129 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 27 T + 435 T^{2} + 4381 T^{3} + 435 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 9 T + 99 T^{2} - 403 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 5 T + 75 T^{2} + 917 T^{3} + 75 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 109 T^{2} + 1365 T^{3} + 109 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 11 T + 193 T^{2} + 1677 T^{3} + 193 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 7 T + 207 T^{2} - 1057 T^{3} + 207 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−11.68876898935071458998037617008, −11.46927784487043750239685597641, −11.30646866607090638627281997410, −11.09985480453711155356091186659, −10.40712211024041910491450461924, −10.33462678075215431562594971829, −10.02917266805030379807198096797, −9.550834590137455943765683279258, −9.161774202653594864080465650357, −8.973815934188217580884209870242, −8.296094328891065865091021257589, −8.176587484609426278270266087853, −8.082682657034107570213469756967, −7.51191873285717262157202617450, −7.50270959552323509370139984132, −6.52528380006161753471808993267, −6.16677339140566145731511886704, −5.77643703948707832010929917193, −5.61737740930925729217237522524, −4.84691216565701289778282224049, −4.52958182550039471163474635583, −4.18529951719078164919223936719, −3.25431298166008658878705217774, −3.17175749010472632307223793978, −2.27315991722013939124163957551, 0, 0, 0,
2.27315991722013939124163957551, 3.17175749010472632307223793978, 3.25431298166008658878705217774, 4.18529951719078164919223936719, 4.52958182550039471163474635583, 4.84691216565701289778282224049, 5.61737740930925729217237522524, 5.77643703948707832010929917193, 6.16677339140566145731511886704, 6.52528380006161753471808993267, 7.50270959552323509370139984132, 7.51191873285717262157202617450, 8.082682657034107570213469756967, 8.176587484609426278270266087853, 8.296094328891065865091021257589, 8.973815934188217580884209870242, 9.161774202653594864080465650357, 9.550834590137455943765683279258, 10.02917266805030379807198096797, 10.33462678075215431562594971829, 10.40712211024041910491450461924, 11.09985480453711155356091186659, 11.30646866607090638627281997410, 11.46927784487043750239685597641, 11.68876898935071458998037617008
