| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s − 5·9-s + 2·10-s + 4·13-s + 16-s + 6·17-s + 5·18-s − 2·20-s + 3·25-s − 4·26-s − 6·29-s − 32-s − 6·34-s − 5·36-s − 14·37-s + 2·40-s + 12·41-s + 10·45-s + 11·49-s − 3·50-s + 4·52-s − 6·53-s + 6·58-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s − 5/3·9-s + 0.632·10-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 1.17·18-s − 0.447·20-s + 3/5·25-s − 0.784·26-s − 1.11·29-s − 0.176·32-s − 1.02·34-s − 5/6·36-s − 2.30·37-s + 0.316·40-s + 1.87·41-s + 1.49·45-s + 11/7·49-s − 0.424·50-s + 0.554·52-s − 0.824·53-s + 0.787·58-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7326532293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7326532293\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 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
−9.422908134669636162293514884732, −8.996361573480265218453634288731, −8.544737917210983194690740059198, −8.285662434899332908926558240119, −7.56981196469027501153188760806, −7.42866571243096715516314299317, −6.61479287984975987379606558452, −5.84669947312883121809728720280, −5.73876439085829035805370640147, −5.01647058733648301862383016754, −4.01173135747517138761270457828, −3.40544989879851656879800288228, −3.06066609883175896466733159370, −1.97170842587956421114470880781, −0.70791386076572374446358125906,
0.70791386076572374446358125906, 1.97170842587956421114470880781, 3.06066609883175896466733159370, 3.40544989879851656879800288228, 4.01173135747517138761270457828, 5.01647058733648301862383016754, 5.73876439085829035805370640147, 5.84669947312883121809728720280, 6.61479287984975987379606558452, 7.42866571243096715516314299317, 7.56981196469027501153188760806, 8.285662434899332908926558240119, 8.544737917210983194690740059198, 8.996361573480265218453634288731, 9.422908134669636162293514884732
