| L(s) = 1 | + 10·3-s − 8·4-s + 73·9-s − 80·12-s + 64·16-s − 212·19-s − 250·25-s + 460·27-s − 584·36-s + 580·43-s + 640·48-s + 686·49-s − 2.12e3·57-s − 512·64-s − 140·67-s − 860·73-s − 2.50e3·75-s + 1.69e3·76-s + 2.62e3·81-s + 3.82e3·97-s + 2.00e3·100-s − 3.68e3·108-s − 2.33e3·121-s + 127-s + 5.80e3·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.92·3-s − 4-s + 2.70·9-s − 1.92·12-s + 16-s − 2.55·19-s − 2·25-s + 3.27·27-s − 2.70·36-s + 2.05·43-s + 1.92·48-s + 2·49-s − 4.92·57-s − 64-s − 0.255·67-s − 1.37·73-s − 3.84·75-s + 2.55·76-s + 3.60·81-s + 3.99·97-s + 2·100-s − 3.27·108-s − 1.75·121-s + 0.000698·127-s + 3.95·129-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.761030658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.761030658\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| 3 | $C_2$ | \( 1 - 10 T + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 18 T + p^{3} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )( 1 + 90 T + p^{3} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 106 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 522 T + p^{3} T^{2} )( 1 + 522 T + p^{3} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 290 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 846 T + p^{3} T^{2} )( 1 + 846 T + p^{3} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 430 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 1350 T + p^{3} T^{2} )( 1 + 1350 T + p^{3} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 1026 T + p^{3} T^{2} )( 1 + 1026 T + p^{3} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 1910 T + p^{3} T^{2} )^{2} \) |
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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
−17.49537251847867538030445534778, −17.17707662552019832708411484596, −16.09487684216355360453776254408, −15.34802142870530138950796854847, −15.02070220758083529490572384490, −14.23141439033694605645255625899, −13.97190630998819148280329432401, −13.01653052415009321306624471507, −12.99333616926732856934058342174, −12.01523309172515279372958118806, −10.54909112750273714796651660759, −10.12289544376962637171298744557, −9.108404416454209027254155882530, −8.873267286634214673132337974018, −8.045852476982021650881049614039, −7.44201540410787076520053969815, −6.07442499956795941981547089926, −4.37526749996133579850317243106, −3.85775137198782991165286996682, −2.26027663966447729927884215960,
2.26027663966447729927884215960, 3.85775137198782991165286996682, 4.37526749996133579850317243106, 6.07442499956795941981547089926, 7.44201540410787076520053969815, 8.045852476982021650881049614039, 8.873267286634214673132337974018, 9.108404416454209027254155882530, 10.12289544376962637171298744557, 10.54909112750273714796651660759, 12.01523309172515279372958118806, 12.99333616926732856934058342174, 13.01653052415009321306624471507, 13.97190630998819148280329432401, 14.23141439033694605645255625899, 15.02070220758083529490572384490, 15.34802142870530138950796854847, 16.09487684216355360453776254408, 17.17707662552019832708411484596, 17.49537251847867538030445534778
