| L(s) = 1 | − 4·2-s − 2·3-s + 8·4-s + 8·6-s − 8·8-s + 2·9-s + 6·11-s − 16·12-s + 4·13-s − 4·16-s + 8·17-s − 8·18-s + 6·19-s − 24·22-s + 16·24-s − 16·26-s + 4·27-s + 8·29-s + 4·31-s + 32·32-s − 12·33-s − 32·34-s + 16·36-s − 16·37-s − 24·38-s − 8·39-s + 4·43-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.15·3-s + 4·4-s + 3.26·6-s − 2.82·8-s + 2/3·9-s + 1.80·11-s − 4.61·12-s + 1.10·13-s − 16-s + 1.94·17-s − 1.88·18-s + 1.37·19-s − 5.11·22-s + 3.26·24-s − 3.13·26-s + 0.769·27-s + 1.48·29-s + 0.718·31-s + 5.65·32-s − 2.08·33-s − 5.48·34-s + 8/3·36-s − 2.63·37-s − 3.89·38-s − 1.28·39-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5931447007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5931447007\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 58 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 60 T^{3} + 199 T^{4} - 60 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 28 T^{3} - 302 T^{4} + 28 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 108 T^{3} + 647 T^{4} - 108 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1338 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 752 T^{3} + 4318 T^{4} + 752 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 57 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 4 p T^{3} - 2 p^{2} T^{4} + 4 p^{2} T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 + 1438 T^{4} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 360 T^{3} + 3854 T^{4} - 360 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 108 T^{3} - 4738 T^{4} + 108 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 30 T + 450 T^{2} + 5220 T^{3} + 49103 T^{4} + 5220 p T^{5} + 450 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 18214 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 70 T^{2} + 7483 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 148 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 22 T + 242 T^{2} - 3036 T^{3} + 35063 T^{4} - 3036 p T^{5} + 242 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 86 T^{2} + 3435 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 150 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.379962157056382291436883757890, −7.74688146716074226541689682474, −7.73760545097658337390851717339, −7.64410973305022098086438252038, −7.24254328040066159019561178917, −7.07412179564542852607016486749, −6.75373015284628119856514899684, −6.52864255387290912487410046196, −6.47362833642957498703992632024, −5.84955239770017212112471238498, −5.78441480650827764211035425254, −5.64290144240409760875496240929, −5.07635353255458939204698659288, −4.78067929007260748573436189825, −4.72069258549140689406239649492, −3.94836274476933997587892634598, −3.90692434001507114955212768179, −3.73068976830149818420888343179, −3.00081411423370612210622339578, −2.83111584436372240333552056504, −2.16761262225022100877252567896, −1.73709995370689421958360416190, −1.02974067913352657908585656083, −0.984444985671682020464278245201, −0.906736594402918891811455148430,
0.906736594402918891811455148430, 0.984444985671682020464278245201, 1.02974067913352657908585656083, 1.73709995370689421958360416190, 2.16761262225022100877252567896, 2.83111584436372240333552056504, 3.00081411423370612210622339578, 3.73068976830149818420888343179, 3.90692434001507114955212768179, 3.94836274476933997587892634598, 4.72069258549140689406239649492, 4.78067929007260748573436189825, 5.07635353255458939204698659288, 5.64290144240409760875496240929, 5.78441480650827764211035425254, 5.84955239770017212112471238498, 6.47362833642957498703992632024, 6.52864255387290912487410046196, 6.75373015284628119856514899684, 7.07412179564542852607016486749, 7.24254328040066159019561178917, 7.64410973305022098086438252038, 7.73760545097658337390851717339, 7.74688146716074226541689682474, 8.379962157056382291436883757890
