| L(s) = 1 | − 4-s + 16-s + 2·17-s − 2·19-s + 2·23-s + 2·47-s + 4·53-s + 2·61-s − 64-s − 2·68-s + 2·76-s − 2·92-s + 2·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s + 2·17-s − 2·19-s + 2·23-s + 2·47-s + 4·53-s + 2·61-s − 64-s − 2·68-s + 2·76-s − 2·92-s + 2·109-s − 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.310510870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310510870\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 53 | $C_1$ | \( ( 1 - T )^{4} \) |
| 59 | $C_2^2$ | \( 1 + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−8.924840331713877732469922041288, −8.526233545765959870534761029644, −8.237971496252668051365799766698, −8.012384872896461317751539076957, −7.32857858289337515784577252081, −7.09624950196721262391253256446, −6.89490832464134106515289714728, −6.22559801106408865588512433282, −5.74601958566405836817432535151, −5.46057489774343533994656622524, −5.32900669016639210065358322191, −4.65177911743131635148143819386, −4.34747779460662295910268534792, −3.83710651073430088809657606669, −3.63399016935119989243603171591, −3.05542926507876727353684843455, −2.48480195480771585385219921642, −2.10842343207064511430585286330, −1.03148092965203080579482246530, −0.898819031395489928031866056968,
0.898819031395489928031866056968, 1.03148092965203080579482246530, 2.10842343207064511430585286330, 2.48480195480771585385219921642, 3.05542926507876727353684843455, 3.63399016935119989243603171591, 3.83710651073430088809657606669, 4.34747779460662295910268534792, 4.65177911743131635148143819386, 5.32900669016639210065358322191, 5.46057489774343533994656622524, 5.74601958566405836817432535151, 6.22559801106408865588512433282, 6.89490832464134106515289714728, 7.09624950196721262391253256446, 7.32857858289337515784577252081, 8.012384872896461317751539076957, 8.237971496252668051365799766698, 8.526233545765959870534761029644, 8.924840331713877732469922041288
