L(s) = 1 | − 4·3-s + 4-s + 6·9-s − 4·12-s + 16-s − 10·25-s + 4·27-s − 8·31-s + 6·36-s + 4·37-s − 24·47-s − 4·48-s + 49-s + 12·53-s − 12·59-s + 64-s − 8·67-s + 40·75-s − 37·81-s − 12·89-s + 32·93-s − 20·97-s − 10·100-s − 8·103-s + 4·108-s − 16·111-s + 12·113-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 1/2·4-s + 2·9-s − 1.15·12-s + 1/4·16-s − 2·25-s + 0.769·27-s − 1.43·31-s + 36-s + 0.657·37-s − 3.50·47-s − 0.577·48-s + 1/7·49-s + 1.64·53-s − 1.56·59-s + 1/8·64-s − 0.977·67-s + 4.61·75-s − 4.11·81-s − 1.27·89-s + 3.31·93-s − 2.03·97-s − 100-s − 0.788·103-s + 0.384·108-s − 1.51·111-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23716 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.74827747141930613672865179341, −10.01782448311884311699973230891, −9.765547119459919407856461234632, −8.888226006934558931566258496787, −8.145906800879012290031802042193, −7.57571100088867902110310233811, −6.82397964191468499058519060303, −6.42455331216463290634831658204, −5.76321350617402115136418955935, −5.57928681742950427486583645839, −4.85990751466979898816213085515, −4.09052365620160024762211243831, −3.03039761753313177141617611476, −1.67120553980965395706477292535, 0,
1.67120553980965395706477292535, 3.03039761753313177141617611476, 4.09052365620160024762211243831, 4.85990751466979898816213085515, 5.57928681742950427486583645839, 5.76321350617402115136418955935, 6.42455331216463290634831658204, 6.82397964191468499058519060303, 7.57571100088867902110310233811, 8.145906800879012290031802042193, 8.888226006934558931566258496787, 9.765547119459919407856461234632, 10.01782448311884311699973230891, 10.74827747141930613672865179341