L(s) = 1 | − 3-s + 4-s + 5-s + 9-s − 12-s − 15-s − 3·16-s + 4·17-s + 20-s − 2·25-s − 27-s + 36-s − 4·37-s − 12·41-s + 8·43-s + 45-s + 3·48-s − 7·49-s − 4·51-s + 8·59-s − 60-s − 7·64-s − 8·67-s + 4·68-s + 2·75-s − 3·80-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/2·4-s + 0.447·5-s + 1/3·9-s − 0.288·12-s − 0.258·15-s − 3/4·16-s + 0.970·17-s + 0.223·20-s − 2/5·25-s − 0.192·27-s + 1/6·36-s − 0.657·37-s − 1.87·41-s + 1.21·43-s + 0.149·45-s + 0.433·48-s − 49-s − 0.560·51-s + 1.04·59-s − 0.129·60-s − 7/8·64-s − 0.977·67-s + 0.485·68-s + 0.230·75-s − 0.335·80-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8899103665\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8899103665\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−11.92650335429280363827949720341, −11.40400582954726890995145174500, −10.86545317284561673034688613344, −10.21657329845347327426449130856, −9.825927791305767909601870766303, −9.117135405500064951174816498766, −8.406884710420726100483844261267, −7.65001517912343532935127686469, −6.98246206054068179852838099217, −6.44841945124364554000865732178, −5.69497291084151422845642595426, −5.12456625206896000687069384338, −4.15801363831078542257449591759, −3.06014077196793113173595850965, −1.80293267241452516983333339657,
1.80293267241452516983333339657, 3.06014077196793113173595850965, 4.15801363831078542257449591759, 5.12456625206896000687069384338, 5.69497291084151422845642595426, 6.44841945124364554000865732178, 6.98246206054068179852838099217, 7.65001517912343532935127686469, 8.406884710420726100483844261267, 9.117135405500064951174816498766, 9.825927791305767909601870766303, 10.21657329845347327426449130856, 10.86545317284561673034688613344, 11.40400582954726890995145174500, 11.92650335429280363827949720341