L(s) = 1 | − 2·3-s + 3·9-s + 12·11-s + 8·19-s − 10·25-s − 4·27-s − 24·33-s + 24·41-s + 8·43-s + 49-s − 16·57-s − 16·67-s − 20·73-s + 20·75-s + 5·81-s + 24·83-s + 24·89-s − 20·97-s + 36·99-s + 12·107-s − 12·113-s + 86·121-s − 48·123-s + 127-s − 16·129-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 3.61·11-s + 1.83·19-s − 2·25-s − 0.769·27-s − 4.17·33-s + 3.74·41-s + 1.21·43-s + 1/7·49-s − 2.11·57-s − 1.95·67-s − 2.34·73-s + 2.30·75-s + 5/9·81-s + 2.63·83-s + 2.54·89-s − 2.03·97-s + 3.61·99-s + 1.16·107-s − 1.12·113-s + 7.81·121-s − 4.32·123-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.850707417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.850707417\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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
−9.000844289389290993593497586164, −7.82437442960794093330464733075, −7.58881932943940700511157248554, −7.19234324757373870654113009154, −6.58911559432542935222512700270, −6.18151745230772575654613193139, −5.84091064569873502208467244139, −5.60383328043951209454966346428, −4.46733872516772697970621091271, −4.37831328158321326146946665271, −3.81005665439346412696166766304, −3.36010318341016445358722759890, −2.20954050006078144234673698125, −1.28052617279627540057290389597, −0.985503276739119959400631494839,
0.985503276739119959400631494839, 1.28052617279627540057290389597, 2.20954050006078144234673698125, 3.36010318341016445358722759890, 3.81005665439346412696166766304, 4.37831328158321326146946665271, 4.46733872516772697970621091271, 5.60383328043951209454966346428, 5.84091064569873502208467244139, 6.18151745230772575654613193139, 6.58911559432542935222512700270, 7.19234324757373870654113009154, 7.58881932943940700511157248554, 7.82437442960794093330464733075, 9.000844289389290993593497586164