L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s + 2·9-s − 14-s − 16-s − 5·17-s − 2·18-s + 16·23-s + 2·25-s − 28-s − 5·32-s + 5·34-s − 2·36-s − 12·41-s − 16·46-s + 8·47-s − 6·49-s − 2·50-s + 3·56-s + 2·63-s + 7·64-s + 5·68-s − 8·71-s + 6·72-s − 4·73-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s + 2/3·9-s − 0.267·14-s − 1/4·16-s − 1.21·17-s − 0.471·18-s + 3.33·23-s + 2/5·25-s − 0.188·28-s − 0.883·32-s + 0.857·34-s − 1/3·36-s − 1.87·41-s − 2.35·46-s + 1.16·47-s − 6/7·49-s − 0.282·50-s + 0.400·56-s + 0.251·63-s + 7/8·64-s + 0.606·68-s − 0.949·71-s + 0.707·72-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6349591019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6349591019\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.48486626051573221058446133700, −11.19953721597607471546206277122, −10.53552652794220010700865602014, −10.15886286083063199768837274861, −9.321204099604706871693586625534, −8.884881245546027501212552296865, −8.583490309010381215708710670513, −7.70117974583691909997473476921, −7.00823262499601846665497793774, −6.73859790548450329591049518652, −5.35983583275801305046937056845, −4.83019142841676289571384399502, −4.20074021204333954966211858674, −2.97566938692753315409261641347, −1.41683349306797468746149066883,
1.41683349306797468746149066883, 2.97566938692753315409261641347, 4.20074021204333954966211858674, 4.83019142841676289571384399502, 5.35983583275801305046937056845, 6.73859790548450329591049518652, 7.00823262499601846665497793774, 7.70117974583691909997473476921, 8.583490309010381215708710670513, 8.884881245546027501212552296865, 9.321204099604706871693586625534, 10.15886286083063199768837274861, 10.53552652794220010700865602014, 11.19953721597607471546206277122, 11.48486626051573221058446133700