| L(s) = 1 | + 2-s + 5-s + 7-s − 8-s + 10-s + 2·11-s + 13-s + 14-s − 16-s − 2·19-s + 2·22-s − 23-s + 26-s + 35-s + 4·37-s − 2·38-s − 40-s − 41-s − 46-s − 47-s + 49-s + 2·53-s + 2·55-s − 56-s − 59-s + 64-s + 65-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s + 7-s − 8-s + 10-s + 2·11-s + 13-s + 14-s − 16-s − 2·19-s + 2·22-s − 23-s + 26-s + 35-s + 4·37-s − 2·38-s − 40-s − 41-s − 46-s − 47-s + 49-s + 2·53-s + 2·55-s − 56-s − 59-s + 64-s + 65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.257736198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.257736198\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.862382048326983612749630284985, −8.801166081160006413097852162282, −8.274814560525591601061423390884, −8.052336699540254873645857124222, −7.59676089410365672855070319435, −6.75934181824754414442003166572, −6.63287889688520218080751389765, −6.30395374696762146082758205902, −5.94346148391479568275326754781, −5.70620898377078290387090232166, −5.31123148712477599616525947491, −4.53022784230852668299061356383, −4.36605480183111432885798404319, −3.97443289757123498831039545828, −3.94559236196324303045140028273, −3.06567604083150391989256361889, −2.54605865091015352993352049611, −2.05325035246164575603023477542, −1.57527254847131054805278285823, −1.02918965750779873688277565407,
1.02918965750779873688277565407, 1.57527254847131054805278285823, 2.05325035246164575603023477542, 2.54605865091015352993352049611, 3.06567604083150391989256361889, 3.94559236196324303045140028273, 3.97443289757123498831039545828, 4.36605480183111432885798404319, 4.53022784230852668299061356383, 5.31123148712477599616525947491, 5.70620898377078290387090232166, 5.94346148391479568275326754781, 6.30395374696762146082758205902, 6.63287889688520218080751389765, 6.75934181824754414442003166572, 7.59676089410365672855070319435, 8.052336699540254873645857124222, 8.274814560525591601061423390884, 8.801166081160006413097852162282, 8.862382048326983612749630284985
