L(s) = 1 | + 3·2-s − 3-s + 7·4-s − 5-s − 3·6-s + 3·7-s + 15·8-s − 3·10-s − 7·12-s + 9·13-s + 9·14-s + 15-s + 30·16-s − 2·17-s + 10·19-s − 7·20-s − 3·21-s − 4·23-s − 15·24-s + 27·26-s + 21·28-s + 10·29-s + 3·30-s + 8·31-s + 57·32-s − 6·34-s − 3·35-s + ⋯ |
L(s) = 1 | + 2.12·2-s − 0.577·3-s + 7/2·4-s − 0.447·5-s − 1.22·6-s + 1.13·7-s + 5.30·8-s − 0.948·10-s − 2.02·12-s + 2.49·13-s + 2.40·14-s + 0.258·15-s + 15/2·16-s − 0.485·17-s + 2.29·19-s − 1.56·20-s − 0.654·21-s − 0.834·23-s − 3.06·24-s + 5.29·26-s + 3.96·28-s + 1.85·29-s + 0.547·30-s + 1.43·31-s + 10.0·32-s − 1.02·34-s − 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(12.04853259\) |
\(L(\frac12)\) |
\(\approx\) |
\(12.04853259\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
good | 2 | $C_2^2:C_4$ | \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 11 p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 15 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 9 T + 18 T^{2} + 115 T^{3} - 789 T^{4} + 115 p T^{5} + 18 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 2 T - 13 T^{2} + 20 T^{3} + 341 T^{4} + 20 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 70 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 200 p T^{5} + 31 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 46 p T^{5} + 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 155 p T^{5} - 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 23 T + 208 T^{2} + 961 T^{3} + 3975 T^{4} + 961 p T^{5} + 208 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 3 T - 43 T^{2} - 45 T^{3} + 2116 T^{4} - 45 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 530 p T^{5} + 131 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 3 T - 7 T^{2} + 441 T^{3} + 4900 T^{4} + 441 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 819 p T^{5} + 253 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 6 T - 57 T^{2} - 130 T^{3} + 4761 T^{4} - 130 p T^{5} - 57 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 5 T + 6 T^{2} + 715 T^{3} + 9821 T^{4} + 715 p T^{5} + 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 21 T + 88 T^{2} - 915 T^{3} - 13199 T^{4} - 915 p T^{5} + 88 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 6655 p T^{5} + 537 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.235321082445680556134545230173, −8.101826672997573657269312259242, −7.72623546436009186779480678521, −7.37859679750734524258532826405, −7.16174617959020833457234934033, −6.88360530153483198766165017126, −6.59499183901965007070164906760, −6.59401636727049831138710924878, −6.13425150205134231176233639154, −5.79045300097549793679347253391, −5.69529136386931716985065287513, −5.53715165773940391836273667989, −5.07337895770830217275557492218, −4.77994006720615815586308854130, −4.60851830779525113903064995186, −4.32439820119788798275205486366, −4.24391109747656867894977990437, −3.53235334792978867159417635623, −3.28885919867012159462202789842, −3.14997464555285877804653587573, −3.05223612619032627501002467731, −2.29116944987890974546669924597, −1.51756182927486248169965921219, −1.43607979429484885392370960091, −1.35730264739982931749737324215,
1.35730264739982931749737324215, 1.43607979429484885392370960091, 1.51756182927486248169965921219, 2.29116944987890974546669924597, 3.05223612619032627501002467731, 3.14997464555285877804653587573, 3.28885919867012159462202789842, 3.53235334792978867159417635623, 4.24391109747656867894977990437, 4.32439820119788798275205486366, 4.60851830779525113903064995186, 4.77994006720615815586308854130, 5.07337895770830217275557492218, 5.53715165773940391836273667989, 5.69529136386931716985065287513, 5.79045300097549793679347253391, 6.13425150205134231176233639154, 6.59401636727049831138710924878, 6.59499183901965007070164906760, 6.88360530153483198766165017126, 7.16174617959020833457234934033, 7.37859679750734524258532826405, 7.72623546436009186779480678521, 8.101826672997573657269312259242, 8.235321082445680556134545230173