L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 5-s + 4·6-s + 2·7-s − 4·8-s + 3·9-s + 2·10-s + 3·11-s − 6·12-s − 2·13-s − 4·14-s + 2·15-s + 5·16-s − 7·17-s − 6·18-s + 3·19-s − 3·20-s − 4·21-s − 6·22-s − 3·23-s + 8·24-s + 5·25-s + 4·26-s − 4·27-s + 6·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.447·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 0.632·10-s + 0.904·11-s − 1.73·12-s − 0.554·13-s − 1.06·14-s + 0.516·15-s + 5/4·16-s − 1.69·17-s − 1.41·18-s + 0.688·19-s − 0.670·20-s − 0.872·21-s − 1.27·22-s − 0.625·23-s + 1.63·24-s + 25-s + 0.784·26-s − 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6243085425\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6243085425\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 26 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + T + 60 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 118 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
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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
−10.78868508975187744268244702582, −10.66146478055000204937840010278, −10.21964007909320570154373489479, −9.767772604161815803693236117002, −9.309884670475511043909031425770, −8.742751029092682789085905086951, −8.343518506082386869702031908164, −8.118377903121081729655039409972, −7.39387180731048144549132281965, −6.91026514947932260405007550027, −6.56221015366234753717816573741, −6.39870794343943751494465686215, −5.47193319690133672490048174894, −5.05266373541693740873949136430, −4.24062375293323472675955780464, −4.19611378843197887720672721530, −2.83488970863995323177745930789, −2.35289732891516897669516912027, −1.28606116306891868803211902621, −0.71120220211023209380963073748,
0.71120220211023209380963073748, 1.28606116306891868803211902621, 2.35289732891516897669516912027, 2.83488970863995323177745930789, 4.19611378843197887720672721530, 4.24062375293323472675955780464, 5.05266373541693740873949136430, 5.47193319690133672490048174894, 6.39870794343943751494465686215, 6.56221015366234753717816573741, 6.91026514947932260405007550027, 7.39387180731048144549132281965, 8.118377903121081729655039409972, 8.343518506082386869702031908164, 8.742751029092682789085905086951, 9.309884670475511043909031425770, 9.767772604161815803693236117002, 10.21964007909320570154373489479, 10.66146478055000204937840010278, 10.78868508975187744268244702582