| L(s) = 1 | + 5.46e3·2-s − 1.29e6·3-s − 3.72e6·4-s − 2.23e8·5-s − 7.06e9·6-s − 2.03e11·8-s + 8.24e11·9-s − 1.22e12·10-s + 1.88e13·11-s + 4.81e12·12-s − 8.69e13·13-s + 2.89e14·15-s − 9.87e14·16-s − 4.52e13·17-s + 4.50e15·18-s − 3.96e15·19-s + 8.32e14·20-s + 1.02e17·22-s − 5.77e16·23-s + 2.63e17·24-s − 2.47e17·25-s − 4.74e17·26-s + 2.91e16·27-s + 1.05e18·29-s + 1.58e18·30-s − 8.29e18·31-s + 1.43e18·32-s + ⋯ |
| L(s) = 1 | + 0.942·2-s − 1.40·3-s − 0.110·4-s − 0.409·5-s − 1.32·6-s − 1.04·8-s + 0.973·9-s − 0.386·10-s + 1.81·11-s + 0.155·12-s − 1.03·13-s + 0.575·15-s − 0.876·16-s − 0.0188·17-s + 0.917·18-s − 0.411·19-s + 0.0454·20-s + 1.70·22-s − 0.549·23-s + 1.47·24-s − 0.832·25-s − 0.975·26-s + 0.0374·27-s + 0.552·29-s + 0.542·30-s − 1.89·31-s + 0.220·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.4841851603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4841851603\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 5.46e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.29e6T + 8.47e11T^{2} \) |
| 5 | \( 1 + 2.23e8T + 2.98e17T^{2} \) |
| 11 | \( 1 - 1.88e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 8.69e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 4.52e13T + 5.77e30T^{2} \) |
| 19 | \( 1 + 3.96e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 5.77e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.05e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 8.29e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 6.11e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 9.27e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.91e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.81e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 5.27e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.55e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 1.62e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 5.66e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 6.75e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.14e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 3.76e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.26e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.64e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 4.68e24T + 4.66e49T^{2} \) |
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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
−11.47735248231436385569249502756, −9.994568222687812818137613787537, −8.809386573683970026678254571356, −7.03051995385775672274515459342, −6.18041296812959769975334886616, −5.23250108418854624765536974438, −4.34116018311604778579623447202, −3.50426724507219163770498543203, −1.69688416689267854927811822205, −0.27246362630494309611095569493,
0.27246362630494309611095569493, 1.69688416689267854927811822205, 3.50426724507219163770498543203, 4.34116018311604778579623447202, 5.23250108418854624765536974438, 6.18041296812959769975334886616, 7.03051995385775672274515459342, 8.809386573683970026678254571356, 9.994568222687812818137613787537, 11.47735248231436385569249502756
