L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s + 16·6-s − 8·8-s − 2·9-s + 4·11-s − 32·12-s − 4·16-s + 8·18-s + 8·19-s − 16·22-s + 32·24-s + 40·27-s + 32·32-s − 16·33-s − 16·36-s − 32·38-s + 24·41-s + 32·44-s + 16·48-s − 160·54-s − 32·57-s + 32·59-s − 64·64-s + 64·66-s + 12·67-s + ⋯ |
L(s) = 1 | − 2.82·2-s − 2.30·3-s + 4·4-s + 6.53·6-s − 2.82·8-s − 2/3·9-s + 1.20·11-s − 9.23·12-s − 16-s + 1.88·18-s + 1.83·19-s − 3.41·22-s + 6.53·24-s + 7.69·27-s + 5.65·32-s − 2.78·33-s − 8/3·36-s − 5.19·38-s + 3.74·41-s + 4.82·44-s + 2.30·48-s − 21.7·54-s − 4.23·57-s + 4.16·59-s − 8·64-s + 7.87·66-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08062396412\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08062396412\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 5 | $C_2^3$ | \( 1 - 41 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 97 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^3$ | \( 1 + 983 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 2143 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 9457 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 132 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−10.29775359764292976226402564992, −9.948812961533655136122455790944, −9.558937967384012206759723928695, −9.296557111346423641802637298505, −9.039021026010379074162872165601, −8.937369428129476185704379742990, −8.431500547116904692045048324818, −8.401852584101365089044823613797, −7.975070347894927258528922692891, −7.71004025179383689216952044764, −7.20390331132935476275839629810, −7.03501935581057867994513552218, −6.64000696166601176652651517608, −6.23515471476455278429499283065, −6.18917970341654490577845454008, −5.52985957166114308155843123738, −5.36789693522002550494786699393, −5.33503796670016970463303208932, −4.53019645201987889968635048287, −4.20778901756617813732786902847, −3.46072327223121112782592454106, −2.57504569506276390440020648449, −2.56799555875125013026098134348, −1.08173189395811207877629341027, −0.69899978609353899100281937804,
0.69899978609353899100281937804, 1.08173189395811207877629341027, 2.56799555875125013026098134348, 2.57504569506276390440020648449, 3.46072327223121112782592454106, 4.20778901756617813732786902847, 4.53019645201987889968635048287, 5.33503796670016970463303208932, 5.36789693522002550494786699393, 5.52985957166114308155843123738, 6.18917970341654490577845454008, 6.23515471476455278429499283065, 6.64000696166601176652651517608, 7.03501935581057867994513552218, 7.20390331132935476275839629810, 7.71004025179383689216952044764, 7.975070347894927258528922692891, 8.401852584101365089044823613797, 8.431500547116904692045048324818, 8.937369428129476185704379742990, 9.039021026010379074162872165601, 9.296557111346423641802637298505, 9.558937967384012206759723928695, 9.948812961533655136122455790944, 10.29775359764292976226402564992