L(s) = 1 | − 5·2-s + 17·4-s − 45·8-s + 89·16-s + 250·25-s − 332·29-s − 85·32-s − 900·37-s − 343·49-s − 1.25e3·50-s − 1.18e3·53-s + 1.66e3·58-s − 287·64-s + 4.50e3·74-s + 1.71e3·98-s + 4.25e3·100-s + 5.90e3·106-s + 108·109-s + 1.34e3·113-s − 5.64e3·116-s + 1.96e3·121-s + 127-s + 2.11e3·128-s + 131-s + 137-s + 139-s − 1.53e4·148-s + ⋯ |
L(s) = 1 | − 1.76·2-s + 17/8·4-s − 1.98·8-s + 1.39·16-s + 2·25-s − 2.12·29-s − 0.469·32-s − 3.99·37-s − 49-s − 3.53·50-s − 3.05·53-s + 3.75·58-s − 0.560·64-s + 7.06·74-s + 1.76·98-s + 17/4·100-s + 5.40·106-s + 0.0949·109-s + 1.11·113-s − 4.51·116-s + 1.47·121-s + 0.000698·127-s + 1.46·128-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 8.49·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4536332152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4536332152\) |
\(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + 5 T + p^{3} T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{3} T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )( 1 + 68 T + p^{3} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 40 T + p^{3} T^{2} )( 1 + 40 T + p^{3} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 166 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 450 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 180 T + p^{3} T^{2} )( 1 + 180 T + p^{3} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 590 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 740 T + p^{3} T^{2} )( 1 + 740 T + p^{3} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 688 T + p^{3} T^{2} )( 1 + 688 T + p^{3} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1384 T + p^{3} T^{2} )( 1 + 1384 T + p^{3} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
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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
−11.63250845780599637163960697604, −11.05492851383318895144459211919, −10.78700854169489903459336827058, −10.46430477521529128115785484782, −9.797904296815373446761075851128, −9.353687475414259363274825069655, −9.028433412611045027406713014986, −8.392659211479266124667662446050, −8.227148943344590358752722757210, −7.30924591156491555564271890825, −7.14949168286708015268952232180, −6.61999438487460971597940687937, −5.97596687320557144716823733695, −5.20910868986212743623347318663, −4.68731400372391173328406912804, −3.45945372621218328865467890163, −3.10933723748747777874024968018, −1.91420099927968933764265093089, −1.55479683112372824848673002048, −0.35574941328729113819783562134,
0.35574941328729113819783562134, 1.55479683112372824848673002048, 1.91420099927968933764265093089, 3.10933723748747777874024968018, 3.45945372621218328865467890163, 4.68731400372391173328406912804, 5.20910868986212743623347318663, 5.97596687320557144716823733695, 6.61999438487460971597940687937, 7.14949168286708015268952232180, 7.30924591156491555564271890825, 8.227148943344590358752722757210, 8.392659211479266124667662446050, 9.028433412611045027406713014986, 9.353687475414259363274825069655, 9.797904296815373446761075851128, 10.46430477521529128115785484782, 10.78700854169489903459336827058, 11.05492851383318895144459211919, 11.63250845780599637163960697604