| L(s) = 1 | + 4·2-s − 4·3-s + 8·4-s − 4·5-s − 16·6-s − 4·7-s + 12·8-s + 4·9-s − 16·10-s + 4·11-s − 32·12-s − 16·14-s + 16·15-s + 15·16-s + 16·18-s − 8·19-s − 32·20-s + 16·21-s + 16·22-s + 12·23-s − 48·24-s + 10·25-s + 4·27-s − 32·28-s + 8·29-s + 64·30-s + 12·31-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 4·4-s − 1.78·5-s − 6.53·6-s − 1.51·7-s + 4.24·8-s + 4/3·9-s − 5.05·10-s + 1.20·11-s − 9.23·12-s − 4.27·14-s + 4.13·15-s + 15/4·16-s + 3.77·18-s − 1.83·19-s − 7.15·20-s + 3.49·21-s + 3.41·22-s + 2.50·23-s − 9.79·24-s + 2·25-s + 0.769·27-s − 6.04·28-s + 1.48·29-s + 11.6·30-s + 2.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.529233535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529233535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T + p^{3} T^{2} - 3 p^{2} T^{3} + 17 T^{4} - 3 p^{3} T^{5} + p^{5} T^{6} - p^{5} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 4 T + 4 p T^{2} + 28 T^{3} + 56 T^{4} + 28 p T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 4 T + 6 T^{2} + 4 T^{3} + 2 T^{4} + 4 p T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 12 T^{2} + 44 T^{3} + 120 T^{4} + 44 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} + 4 p T^{3} - 168 T^{4} + 4 p^{2} T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 184 T^{3} + 1042 T^{4} + 184 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 68 T^{2} - 316 T^{3} + 1496 T^{4} - 316 p T^{5} + 68 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 18 T^{2} + 120 T^{3} - 926 T^{4} + 120 p T^{5} + 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 36 T^{2} + 12 p T^{3} - 3816 T^{4} + 12 p^{2} T^{5} + 36 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 20 T + 150 T^{2} + 500 T^{3} + 1250 T^{4} + 500 p T^{5} + 150 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 376 T^{3} + 4402 T^{4} - 376 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2854 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 150 T^{2} + 500 T^{3} + 1250 T^{4} + 500 p T^{5} + 150 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 20 T + 300 T^{2} - 3420 T^{3} + 33400 T^{4} - 3420 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 672 T^{3} + 4802 T^{4} + 672 p T^{5} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 12 T + 68 T^{2} + 540 T^{3} + 4184 T^{4} + 540 p T^{5} + 68 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1264 T^{3} + 12466 T^{4} + 1264 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 27874 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T + 54 T^{2} - 1388 T^{3} - 4894 T^{4} - 1388 p T^{5} + 54 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.524532791187516612344433879599, −8.162166975538405952531155189865, −8.161762985081035208484906279379, −7.62001244816335549552870937321, −6.99980190141439052744357572349, −6.97200223690991962723960273099, −6.90504821037325140677386472411, −6.70418953551812336579180399846, −6.27782923678038741122707353322, −6.05440500560575824873729591688, −6.04541997972059184293274876409, −5.56354563964682452666082044251, −5.24748588045161929235453160516, −5.05760924249245720573450161156, −4.81407233438071050570258180512, −4.42807080742808074278823104632, −4.24151557366026255685007611695, −4.01424661352667227821488976526, −3.89994707217805160076139320815, −3.11406242110723204309625503692, −3.02379399210395561506730242619, −2.96098701911479966556842595915, −2.20554983908297703593287714541, −1.11634992736315779072119160441, −0.50129655905508413860165509902,
0.50129655905508413860165509902, 1.11634992736315779072119160441, 2.20554983908297703593287714541, 2.96098701911479966556842595915, 3.02379399210395561506730242619, 3.11406242110723204309625503692, 3.89994707217805160076139320815, 4.01424661352667227821488976526, 4.24151557366026255685007611695, 4.42807080742808074278823104632, 4.81407233438071050570258180512, 5.05760924249245720573450161156, 5.24748588045161929235453160516, 5.56354563964682452666082044251, 6.04541997972059184293274876409, 6.05440500560575824873729591688, 6.27782923678038741122707353322, 6.70418953551812336579180399846, 6.90504821037325140677386472411, 6.97200223690991962723960273099, 6.99980190141439052744357572349, 7.62001244816335549552870937321, 8.161762985081035208484906279379, 8.162166975538405952531155189865, 8.524532791187516612344433879599
