| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s + 4·13-s + 16-s − 3·18-s + 16·23-s − 6·25-s + 4·26-s + 32-s − 3·36-s + 20·37-s + 16·46-s − 16·47-s − 14·49-s − 6·50-s + 4·52-s − 24·59-s − 12·61-s + 64-s + 16·71-s − 3·72-s + 20·73-s + 20·74-s + 9·81-s + 16·83-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s + 1.10·13-s + 1/4·16-s − 0.707·18-s + 3.33·23-s − 6/5·25-s + 0.784·26-s + 0.176·32-s − 1/2·36-s + 3.28·37-s + 2.35·46-s − 2.33·47-s − 2·49-s − 0.848·50-s + 0.554·52-s − 3.12·59-s − 1.53·61-s + 1/8·64-s + 1.89·71-s − 0.353·72-s + 2.34·73-s + 2.32·74-s + 81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276768 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276768 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.838960258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.838960258\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.056498267458590117851162621111, −8.228150792954604791756150599674, −7.78432113554336725186102006286, −7.76857292337562945056886541715, −6.68841524677704446114152278891, −6.32858579420662158347624019611, −6.20957515971754775976198016729, −5.43138638838697497763428859213, −4.76892395639548888503622113417, −4.74203912193695697715499891246, −3.68575806630196919369820756911, −3.16188554299090604324575557253, −2.90212986037219285929538328410, −1.88722882510241810597714442566, −0.944147784971163780613450180079,
0.944147784971163780613450180079, 1.88722882510241810597714442566, 2.90212986037219285929538328410, 3.16188554299090604324575557253, 3.68575806630196919369820756911, 4.74203912193695697715499891246, 4.76892395639548888503622113417, 5.43138638838697497763428859213, 6.20957515971754775976198016729, 6.32858579420662158347624019611, 6.68841524677704446114152278891, 7.76857292337562945056886541715, 7.78432113554336725186102006286, 8.228150792954604791756150599674, 9.056498267458590117851162621111
