| L(s) = 1 | − 4-s − 4·13-s − 3·16-s − 10·19-s − 7·25-s − 10·31-s − 14·37-s − 8·43-s + 4·52-s − 16·61-s + 7·64-s + 28·67-s + 8·73-s + 10·76-s + 16·79-s + 8·97-s + 7·100-s − 10·103-s − 14·109-s − 19·121-s + 10·124-s + 127-s + 131-s + 137-s + 139-s + 14·148-s + 149-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.10·13-s − 3/4·16-s − 2.29·19-s − 7/5·25-s − 1.79·31-s − 2.30·37-s − 1.21·43-s + 0.554·52-s − 2.04·61-s + 7/8·64-s + 3.42·67-s + 0.936·73-s + 1.14·76-s + 1.80·79-s + 0.812·97-s + 7/10·100-s − 0.985·103-s − 1.34·109-s − 1.72·121-s + 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.15·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−9.229471242417547405326178888526, −9.150601326935250268842732010899, −8.687424163269057293029812961513, −8.247548082818164878249800608539, −7.82087867699404347933620975173, −7.50745657439264564636621318103, −6.77934111258128056454303642378, −6.64049181973731317540338752503, −6.28900794618881628796059323236, −5.46304339025497881985208712235, −5.12004614668373612663990118261, −4.92881054200784352642078274256, −4.14683737870221036945801116423, −3.84416613660774468741132232479, −3.48172689855521382668246180831, −2.44202828862067332005023082242, −2.15760110786234079539525541640, −1.64011161567023595614245826782, 0, 0,
1.64011161567023595614245826782, 2.15760110786234079539525541640, 2.44202828862067332005023082242, 3.48172689855521382668246180831, 3.84416613660774468741132232479, 4.14683737870221036945801116423, 4.92881054200784352642078274256, 5.12004614668373612663990118261, 5.46304339025497881985208712235, 6.28900794618881628796059323236, 6.64049181973731317540338752503, 6.77934111258128056454303642378, 7.50745657439264564636621318103, 7.82087867699404347933620975173, 8.247548082818164878249800608539, 8.687424163269057293029812961513, 9.150601326935250268842732010899, 9.229471242417547405326178888526
