L(s) = 1 | − 2·3-s − 2·4-s − 5·9-s + 4·12-s − 2·13-s + 4·16-s − 62·19-s − 22·25-s + 28·27-s − 14·31-s + 10·36-s − 2·37-s + 4·39-s − 62·43-s − 8·48-s + 4·52-s + 124·57-s + 100·61-s − 8·64-s + 130·67-s − 194·73-s + 44·75-s + 124·76-s − 206·79-s − 11·81-s + 28·93-s − 332·97-s + ⋯ |
L(s) = 1 | − 2/3·3-s − 1/2·4-s − 5/9·9-s + 1/3·12-s − 0.153·13-s + 1/4·16-s − 3.26·19-s − 0.879·25-s + 1.03·27-s − 0.451·31-s + 5/18·36-s − 0.0540·37-s + 4/39·39-s − 1.44·43-s − 1/6·48-s + 1/13·52-s + 2.17·57-s + 1.63·61-s − 1/8·64-s + 1.94·67-s − 2.65·73-s + 0.586·75-s + 1.63·76-s − 2.60·79-s − 0.135·81-s + 0.301·93-s − 3.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3664326912\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3664326912\) |
\(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 22 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 506 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 986 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1394 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2210 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2618 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4970 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6890 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6554 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 103 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 11978 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1730 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 166 T + p^{2} T^{2} )^{2} \) |
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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
−11.54099593810411389838942966422, −11.42246508583110982931956675307, −10.92546721175649204149241301385, −10.35917386067993420639895914539, −10.02675497105011586656715882800, −9.559721191266592519689023540103, −8.639924403211569938747714019135, −8.433890274118731173033881026709, −8.390217434238413612560123698455, −7.23967137742918327717320971194, −6.92873019094949383832012862066, −6.09308076735885301058227264457, −6.00126048529382743648643433039, −5.27471474430816613011688006242, −4.57936349551115430058826044095, −4.20952292335642111774566553367, −3.47280463764752374706074937287, −2.50449225655717400743056059183, −1.78804504103841022946141001341, −0.30063623931840240504964648974,
0.30063623931840240504964648974, 1.78804504103841022946141001341, 2.50449225655717400743056059183, 3.47280463764752374706074937287, 4.20952292335642111774566553367, 4.57936349551115430058826044095, 5.27471474430816613011688006242, 6.00126048529382743648643433039, 6.09308076735885301058227264457, 6.92873019094949383832012862066, 7.23967137742918327717320971194, 8.390217434238413612560123698455, 8.433890274118731173033881026709, 8.639924403211569938747714019135, 9.559721191266592519689023540103, 10.02675497105011586656715882800, 10.35917386067993420639895914539, 10.92546721175649204149241301385, 11.42246508583110982931956675307, 11.54099593810411389838942966422