| L(s) = 1 | + 2·3-s + 2·7-s − 3·9-s + 16·19-s + 4·21-s − 25-s − 14·27-s + 10·31-s − 2·37-s − 3·49-s − 12·53-s + 32·57-s + 6·59-s − 6·63-s − 2·75-s − 4·81-s + 12·83-s + 20·93-s + 16·103-s + 4·109-s − 4·111-s − 30·113-s + 121-s + 127-s + 131-s + 32·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s − 9-s + 3.67·19-s + 0.872·21-s − 1/5·25-s − 2.69·27-s + 1.79·31-s − 0.328·37-s − 3/7·49-s − 1.64·53-s + 4.23·57-s + 0.781·59-s − 0.755·63-s − 0.230·75-s − 4/9·81-s + 1.31·83-s + 2.07·93-s + 1.57·103-s + 0.383·109-s − 0.379·111-s − 2.82·113-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.936582794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.936582794\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.607074551585026726901312251020, −8.132665807481574488057539990847, −7.87110218888344770753210710353, −7.60852684015633966980596452192, −7.00173382186013135401075978172, −6.30575248548466868036234960594, −5.78408513013589990534397837958, −5.14377144773301545133518979753, −5.11227132195889762323685903220, −4.20362350660513118934837301951, −3.27381985666804460223319703873, −3.25753016666133195336025247275, −2.66054946243124667318595839863, −1.84385955720565734738536893245, −0.942835376243860514734661121549,
0.942835376243860514734661121549, 1.84385955720565734738536893245, 2.66054946243124667318595839863, 3.25753016666133195336025247275, 3.27381985666804460223319703873, 4.20362350660513118934837301951, 5.11227132195889762323685903220, 5.14377144773301545133518979753, 5.78408513013589990534397837958, 6.30575248548466868036234960594, 7.00173382186013135401075978172, 7.60852684015633966980596452192, 7.87110218888344770753210710353, 8.132665807481574488057539990847, 8.607074551585026726901312251020
