| L(s) = 1 | + 1.06e4·2-s + 4.62e5·3-s + 7.92e7·4-s − 7.26e8·5-s + 4.91e9·6-s + 4.85e11·8-s − 6.33e11·9-s − 7.71e12·10-s + 1.89e13·11-s + 3.66e13·12-s − 1.00e14·13-s − 3.36e14·15-s + 2.49e15·16-s − 4.82e14·17-s − 6.72e15·18-s + 3.41e15·19-s − 5.75e16·20-s + 2.00e17·22-s − 8.11e16·23-s + 2.24e17·24-s + 2.30e17·25-s − 1.06e18·26-s − 6.85e17·27-s + 1.97e17·29-s − 3.57e18·30-s − 6.33e18·31-s + 1.02e19·32-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 0.502·3-s + 2.36·4-s − 1.33·5-s + 0.921·6-s + 2.49·8-s − 0.747·9-s − 2.44·10-s + 1.81·11-s + 1.18·12-s − 1.19·13-s − 0.669·15-s + 2.21·16-s − 0.201·17-s − 1.37·18-s + 0.353·19-s − 3.14·20-s + 3.32·22-s − 0.772·23-s + 1.25·24-s + 0.772·25-s − 2.18·26-s − 0.878·27-s + 0.103·29-s − 1.22·30-s − 1.44·31-s + 1.56·32-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 | \( 1 - 1.06e4T + 3.35e7T^{2} \) |
| 3 | \( 1 - 4.62e5T + 8.47e11T^{2} \) |
| 5 | \( 1 + 7.26e8T + 2.98e17T^{2} \) |
| 11 | \( 1 - 1.89e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.00e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 4.82e14T + 5.77e30T^{2} \) |
| 19 | \( 1 - 3.41e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 8.11e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.97e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 6.33e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 4.34e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 3.51e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.02e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 6.22e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 2.75e20T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.45e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.06e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 3.80e21T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.80e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 4.22e22T + 3.82e46T^{2} \) |
| 79 | \( 1 + 2.63e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 5.25e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 2.06e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 6.63e24T + 4.66e49T^{2} \) |
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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
−11.03629303272187088625767203041, −9.131143562156472251910334162532, −7.70215881169855322517817476196, −6.86492202161999441085497658802, −5.64237772158343765122039574369, −4.37081399989368109816469685312, −3.76259414302824998660549086288, −2.93370681799629074342224446001, −1.73854790954418263793304357499, 0,
1.73854790954418263793304357499, 2.93370681799629074342224446001, 3.76259414302824998660549086288, 4.37081399989368109816469685312, 5.64237772158343765122039574369, 6.86492202161999441085497658802, 7.70215881169855322517817476196, 9.131143562156472251910334162532, 11.03629303272187088625767203041
