| L(s) = 1 | + 5.31e5·3-s − 6.59e8·5-s + 5.77e9·7-s + 2.82e11·9-s + 1.69e13·11-s + 2.46e13·13-s − 3.50e14·15-s − 5.10e14·17-s + 1.76e16·19-s + 3.07e15·21-s + 1.34e17·23-s + 1.37e17·25-s + 1.50e17·27-s − 3.40e18·29-s − 2.54e18·31-s + 9.02e18·33-s − 3.81e18·35-s − 1.75e19·37-s + 1.31e19·39-s − 1.90e20·41-s + 1.16e20·43-s − 1.86e20·45-s − 6.74e20·47-s − 1.30e21·49-s − 2.71e20·51-s + 1.40e21·53-s − 1.11e22·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.20·5-s + 0.157·7-s + 0.333·9-s + 1.63·11-s + 0.293·13-s − 0.697·15-s − 0.212·17-s + 1.82·19-s + 0.0911·21-s + 1.28·23-s + 0.460·25-s + 0.192·27-s − 1.78·29-s − 0.580·31-s + 0.941·33-s − 0.190·35-s − 0.437·37-s + 0.169·39-s − 1.31·41-s + 0.444·43-s − 0.402·45-s − 0.846·47-s − 0.975·49-s − 0.122·51-s + 0.393·53-s − 1.97·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(2.742688485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.742688485\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.31e5T \) |
| good | 5 | \( 1 + 6.59e8T + 2.98e17T^{2} \) |
| 7 | \( 1 - 5.77e9T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.69e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 2.46e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 5.10e14T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.76e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.34e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 3.40e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 2.54e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 1.75e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.90e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.16e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 6.74e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 1.40e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 2.44e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.64e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 2.07e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.15e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 1.69e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 2.79e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 5.68e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.67e23T + 5.42e48T^{2} \) |
| 97 | \( 1 - 1.20e25T + 4.66e49T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.27857764734859013095133423985, −9.562542929420893982590980193143, −8.746289166374876062092888489819, −7.60111523255754678186891631242, −6.81774250907727344696150844231, −5.14376264981108529391410331115, −3.79815625953382801411029409052, −3.38843550612508737514040111106, −1.68875284423134276618370847208, −0.71813340214964862987116786123,
0.71813340214964862987116786123, 1.68875284423134276618370847208, 3.38843550612508737514040111106, 3.79815625953382801411029409052, 5.14376264981108529391410331115, 6.81774250907727344696150844231, 7.60111523255754678186891631242, 8.746289166374876062092888489819, 9.562542929420893982590980193143, 11.27857764734859013095133423985
