| L(s) = 1 | − 3·2-s − 6·3-s + 7·4-s + 4·5-s + 18·6-s + 7·7-s − 15·8-s + 23·9-s − 12·10-s + 2·11-s − 42·12-s − 6·13-s − 21·14-s − 24·15-s + 30·16-s + 8·17-s − 69·18-s + 5·19-s + 28·20-s − 42·21-s − 6·22-s − 16·23-s + 90·24-s − 10·25-s + 18·26-s − 70·27-s + 49·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 3.46·3-s + 7/2·4-s + 1.78·5-s + 7.34·6-s + 2.64·7-s − 5.30·8-s + 23/3·9-s − 3.79·10-s + 0.603·11-s − 12.1·12-s − 1.66·13-s − 5.61·14-s − 6.19·15-s + 15/2·16-s + 1.94·17-s − 16.2·18-s + 1.14·19-s + 6.26·20-s − 9.16·21-s − 1.27·22-s − 3.33·23-s + 18.3·24-s − 2·25-s + 3.53·26-s − 13.4·27-s + 9.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2209964616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2209964616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 31 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + 3 T + p T^{2} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 + 2 p T + 13 T^{2} + 10 T^{3} + T^{4} + 10 p T^{5} + 13 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 7 | $C_2^2:C_4$ | \( 1 - p T + 12 T^{2} + 25 T^{3} - 139 T^{4} + 25 p T^{5} + 12 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - 2 T - 7 T^{2} + 36 T^{3} + 5 T^{4} + 36 p T^{5} - 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 6 T + 3 T^{2} - 10 T^{3} + 81 T^{4} - 10 p T^{5} + 3 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 8 T + 7 T^{2} + 110 T^{3} - 579 T^{4} + 110 p T^{5} + 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 5 T - 4 T^{2} - 25 T^{3} + 481 T^{4} - 25 p T^{5} - 4 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 16 T + 113 T^{2} + 570 T^{3} + 2741 T^{4} + 570 p T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 11 T^{2} + 90 T^{3} + 661 T^{4} + 90 p T^{5} + 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $C_4\times C_2$ | \( 1 + 7 T + 8 T^{2} - 231 T^{3} - 1945 T^{4} - 231 p T^{5} + 8 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 4 T - 27 T^{2} - 110 T^{3} + 2381 T^{4} - 110 p T^{5} - 27 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 380 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 4 T + 43 T^{2} - 380 T^{3} + 4761 T^{4} - 380 p T^{5} + 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 5 T - 44 T^{2} + 25 T^{3} + 3801 T^{4} + 25 p T^{5} - 44 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 27 T + 308 T^{2} + 2499 T^{3} + 20605 T^{4} + 2499 p T^{5} + 308 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 6 T + 3 T^{2} + 640 T^{3} + 9141 T^{4} + 640 p T^{5} + 3 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 10 T + 81 T^{2} - 1190 T^{3} + 16121 T^{4} - 1190 p T^{5} + 81 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 26 T + 193 T^{2} - 380 T^{3} - 13419 T^{4} - 380 p T^{5} + 193 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 10 T + 71 T^{2} + 1290 T^{3} + 19001 T^{4} + 1290 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 13 T + 192 T^{2} + 2195 T^{3} + 31031 T^{4} + 2195 p T^{5} + 192 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.19107823857738046858370642193, −6.96501616088996773242499993358, −6.91061165020689818772153688904, −6.31580091799397707916697241262, −6.27430881569634857592251360642, −5.92904640615407829586086660799, −5.78117193590382041177987668037, −5.64257707302181459522575978525, −5.53390450902449252912483147368, −5.48936158206668730744726690413, −5.32988207904092559748705084839, −4.58859242658491492957725143145, −4.50056322037829163346840013118, −4.47558929895382278533692366460, −3.78863347005517026811438150027, −3.75752485926595317362251617948, −3.35252645039132138228562495756, −2.41474325710135716185396258326, −2.32017742137149622013443105213, −2.08684166337555974650192134456, −1.71015839964685461248855931979, −1.55192706336212336999669580349, −1.25228924235585619892840879494, −0.995985636521937484128552229562, −0.21802807176575979324930805511,
0.21802807176575979324930805511, 0.995985636521937484128552229562, 1.25228924235585619892840879494, 1.55192706336212336999669580349, 1.71015839964685461248855931979, 2.08684166337555974650192134456, 2.32017742137149622013443105213, 2.41474325710135716185396258326, 3.35252645039132138228562495756, 3.75752485926595317362251617948, 3.78863347005517026811438150027, 4.47558929895382278533692366460, 4.50056322037829163346840013118, 4.58859242658491492957725143145, 5.32988207904092559748705084839, 5.48936158206668730744726690413, 5.53390450902449252912483147368, 5.64257707302181459522575978525, 5.78117193590382041177987668037, 5.92904640615407829586086660799, 6.27430881569634857592251360642, 6.31580091799397707916697241262, 6.91061165020689818772153688904, 6.96501616088996773242499993358, 7.19107823857738046858370642193
