| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s + 3·9-s − 4·10-s − 6·13-s − 4·16-s + 6·17-s − 6·18-s + 4·20-s + 3·25-s + 12·26-s − 2·29-s + 8·32-s − 12·34-s + 6·36-s − 12·41-s + 6·45-s − 6·50-s − 12·52-s − 20·53-s + 4·58-s − 8·64-s − 12·65-s + 12·68-s − 12·73-s − 8·80-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s + 9-s − 1.26·10-s − 1.66·13-s − 16-s + 1.45·17-s − 1.41·18-s + 0.894·20-s + 3/5·25-s + 2.35·26-s − 0.371·29-s + 1.41·32-s − 2.05·34-s + 36-s − 1.87·41-s + 0.894·45-s − 0.848·50-s − 1.66·52-s − 2.74·53-s + 0.525·58-s − 64-s − 1.48·65-s + 1.45·68-s − 1.40·73-s − 0.894·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80457391816395721714669401072, −7.56848203557801886905576002740, −7.30703478828022800288631659705, −6.66812515626338690414111791742, −6.46027048767020078983890297601, −5.69074574951383439724683067976, −5.27291442121198189570792311479, −4.68059577608072660097150741909, −4.49258853804675338518966926948, −3.46612148832537381816016895570, −3.00378692647977841566927386749, −2.19920226028059021829079345585, −1.70898778880961935771596537039, −1.18708333743445212285166420152, 0,
1.18708333743445212285166420152, 1.70898778880961935771596537039, 2.19920226028059021829079345585, 3.00378692647977841566927386749, 3.46612148832537381816016895570, 4.49258853804675338518966926948, 4.68059577608072660097150741909, 5.27291442121198189570792311479, 5.69074574951383439724683067976, 6.46027048767020078983890297601, 6.66812515626338690414111791742, 7.30703478828022800288631659705, 7.56848203557801886905576002740, 7.80457391816395721714669401072
