L(s) = 1 | − 4·7-s + 9-s − 4·17-s − 8·23-s + 6·25-s + 12·31-s + 12·41-s − 8·47-s − 2·49-s − 4·63-s − 24·71-s − 20·73-s − 20·79-s + 81-s − 28·89-s + 20·97-s + 20·103-s + 4·113-s + 16·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + ⋯ |
L(s) = 1 | − 1.51·7-s + 1/3·9-s − 0.970·17-s − 1.66·23-s + 6/5·25-s + 2.15·31-s + 1.87·41-s − 1.16·47-s − 2/7·49-s − 0.503·63-s − 2.84·71-s − 2.34·73-s − 2.25·79-s + 1/9·81-s − 2.96·89-s + 2.03·97-s + 1.97·103-s + 0.376·113-s + 1.46·119-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−8.985911500488081519481984584802, −8.649034097305583887898795668373, −8.230854059399453444196153124887, −7.30623361332829545195635303299, −7.27371486188202795087403560747, −6.30170361535159036673173480205, −6.27798985919626463740519529757, −5.81778647559970487461786585347, −4.60632905768716122167488407077, −4.56007962912554840821971374527, −3.77214318657172177768570145308, −2.92708427914109016867182117304, −2.65190583901467307643981603895, −1.43950309180346771366044691508, 0,
1.43950309180346771366044691508, 2.65190583901467307643981603895, 2.92708427914109016867182117304, 3.77214318657172177768570145308, 4.56007962912554840821971374527, 4.60632905768716122167488407077, 5.81778647559970487461786585347, 6.27798985919626463740519529757, 6.30170361535159036673173480205, 7.27371486188202795087403560747, 7.30623361332829545195635303299, 8.230854059399453444196153124887, 8.649034097305583887898795668373, 8.985911500488081519481984584802