| L(s) = 1 | − 7-s − 2·19-s − 2·25-s − 4·31-s + 2·37-s + 2·103-s + 4·109-s − 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 7-s − 2·19-s − 2·25-s − 4·31-s + 2·37-s + 2·103-s + 4·109-s − 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5879557198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5879557198\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.121047979608355700006923208725, −8.630524834160870791336923037494, −8.574731314814365310145791470677, −7.78412561309109524335628426396, −7.60789524748225437968582825859, −7.32970745173223716883012223910, −6.81736301151229187880720108905, −6.30098866653607628324910877047, −6.14898580901511067377874144694, −5.71898973738556147591147785606, −5.46230143823996702612338659992, −4.74674396255946727233850078329, −4.36824646562009501617559499222, −3.78775619543251877696762409812, −3.73762750455394345704976377007, −3.16580983972864031588914220981, −2.48659880096954148029614666009, −1.94461156060035397295233545608, −1.79145107251902452753521301044, −0.44453482659693476430196356370,
0.44453482659693476430196356370, 1.79145107251902452753521301044, 1.94461156060035397295233545608, 2.48659880096954148029614666009, 3.16580983972864031588914220981, 3.73762750455394345704976377007, 3.78775619543251877696762409812, 4.36824646562009501617559499222, 4.74674396255946727233850078329, 5.46230143823996702612338659992, 5.71898973738556147591147785606, 6.14898580901511067377874144694, 6.30098866653607628324910877047, 6.81736301151229187880720108905, 7.32970745173223716883012223910, 7.60789524748225437968582825859, 7.78412561309109524335628426396, 8.574731314814365310145791470677, 8.630524834160870791336923037494, 9.121047979608355700006923208725
