| L(s) = 1 | + 2·5-s + 2·11-s − 2·13-s + 2·17-s − 2·19-s − 2·23-s + 25-s + 2·41-s + 2·43-s + 4·55-s − 4·65-s + 4·85-s − 4·95-s + 2·103-s − 2·107-s − 2·113-s − 4·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 4·143-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2·5-s + 2·11-s − 2·13-s + 2·17-s − 2·19-s − 2·23-s + 25-s + 2·41-s + 2·43-s + 4·55-s − 4·65-s + 4·85-s − 4·95-s + 2·103-s − 2·107-s − 2·113-s − 4·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s − 4·143-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1498176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1498176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.557636638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557636638\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T^{4} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + T^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_2^2$ | \( 1 + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.867407862659683404900532480026, −9.780824532793101687100032429123, −9.394340823552955517760951048873, −9.194040615529656349970865681908, −8.641408709000047708409357887267, −8.044330708331787325220962429865, −7.54386169340081588467820835961, −7.41320994977026549169208484515, −6.46782998805953865523129560527, −6.43600279256790155076679622790, −5.83447815758222056735076990041, −5.80471549177089831131357871858, −5.16388001754828158897653412547, −4.53235417807861078972735299753, −3.89756949744959439524906263284, −3.89187089406014943455718454888, −2.54406568015478086719457527751, −2.47849886880256181301006945127, −1.83283734780885241631386477879, −1.25591108254819659868895857152,
1.25591108254819659868895857152, 1.83283734780885241631386477879, 2.47849886880256181301006945127, 2.54406568015478086719457527751, 3.89187089406014943455718454888, 3.89756949744959439524906263284, 4.53235417807861078972735299753, 5.16388001754828158897653412547, 5.80471549177089831131357871858, 5.83447815758222056735076990041, 6.43600279256790155076679622790, 6.46782998805953865523129560527, 7.41320994977026549169208484515, 7.54386169340081588467820835961, 8.044330708331787325220962429865, 8.641408709000047708409357887267, 9.194040615529656349970865681908, 9.394340823552955517760951048873, 9.780824532793101687100032429123, 9.867407862659683404900532480026
