| L(s) = 1 | + 367.·2-s + 1.08e6·3-s − 3.34e7·4-s + 3.08e8·5-s + 3.96e8·6-s − 2.45e10·8-s + 3.20e11·9-s + 1.13e11·10-s + 1.61e13·11-s − 3.61e13·12-s − 4.66e13·13-s + 3.32e14·15-s + 1.11e15·16-s − 2.51e15·17-s + 1.17e14·18-s − 5.57e15·19-s − 1.02e16·20-s + 5.93e15·22-s + 1.23e17·23-s − 2.65e16·24-s − 2.03e17·25-s − 1.71e16·26-s − 5.69e17·27-s − 2.33e18·29-s + 1.22e17·30-s − 4.77e17·31-s + 1.23e18·32-s + ⋯ |
| L(s) = 1 | + 0.0633·2-s + 1.17·3-s − 0.995·4-s + 0.564·5-s + 0.0744·6-s − 0.126·8-s + 0.377·9-s + 0.0357·10-s + 1.55·11-s − 1.16·12-s − 0.555·13-s + 0.662·15-s + 0.987·16-s − 1.04·17-s + 0.0239·18-s − 0.577·19-s − 0.562·20-s + 0.0984·22-s + 1.17·23-s − 0.148·24-s − 0.681·25-s − 0.0352·26-s − 0.730·27-s − 1.22·29-s + 0.0420·30-s − 0.108·31-s + 0.189·32-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
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 - 367.T + 3.35e7T^{2} \) |
| 3 | \( 1 - 1.08e6T + 8.47e11T^{2} \) |
| 5 | \( 1 - 3.08e8T + 2.98e17T^{2} \) |
| 11 | \( 1 - 1.61e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 4.66e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 2.51e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 5.57e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.23e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 2.33e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.77e17T + 1.92e37T^{2} \) |
| 37 | \( 1 - 4.92e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.27e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 5.02e19T + 6.86e40T^{2} \) |
| 47 | \( 1 - 2.55e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.42e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.07e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 2.90e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 7.44e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.68e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.62e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 7.23e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 9.06e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.74e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 4.24e24T + 4.66e49T^{2} \) |
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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
−9.841918258767672195154597054215, −9.116131346392541699310725452625, −8.570056675218837315957181289596, −7.15995914022425894920756538287, −5.82355880947268901017899879457, −4.43741045444836247904316124975, −3.64126369544419030285777405370, −2.42146384631535454599878547807, −1.38723369024990795425875209715, 0,
1.38723369024990795425875209715, 2.42146384631535454599878547807, 3.64126369544419030285777405370, 4.43741045444836247904316124975, 5.82355880947268901017899879457, 7.15995914022425894920756538287, 8.570056675218837315957181289596, 9.116131346392541699310725452625, 9.841918258767672195154597054215
