| L(s) = 1 | − 3.30e3·2-s − 1.44e6·3-s − 2.26e7·4-s + 7.72e8·5-s + 4.79e9·6-s + 1.85e11·8-s + 1.25e12·9-s − 2.55e12·10-s − 3.84e12·11-s + 3.28e13·12-s + 1.01e14·13-s − 1.12e15·15-s + 1.45e14·16-s + 4.17e15·17-s − 4.14e15·18-s − 7.47e15·19-s − 1.74e16·20-s + 1.27e16·22-s − 2.00e16·23-s − 2.69e17·24-s + 2.99e17·25-s − 3.35e17·26-s − 5.91e17·27-s − 1.87e18·29-s + 3.70e18·30-s − 3.71e18·31-s − 6.71e18·32-s + ⋯ |
| L(s) = 1 | − 0.570·2-s − 1.57·3-s − 0.674·4-s + 1.41·5-s + 0.898·6-s + 0.955·8-s + 1.48·9-s − 0.807·10-s − 0.369·11-s + 1.06·12-s + 1.20·13-s − 2.23·15-s + 0.129·16-s + 1.73·17-s − 0.845·18-s − 0.774·19-s − 0.954·20-s + 0.210·22-s − 0.190·23-s − 1.50·24-s + 1.00·25-s − 0.690·26-s − 0.758·27-s − 0.984·29-s + 1.27·30-s − 0.847·31-s − 1.02·32-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 | \( 1 + 3.30e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.44e6T + 8.47e11T^{2} \) |
| 5 | \( 1 - 7.72e8T + 2.98e17T^{2} \) |
| 11 | \( 1 + 3.84e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.01e14T + 7.05e27T^{2} \) |
| 17 | \( 1 - 4.17e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 7.47e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 2.00e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.87e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 3.71e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 5.79e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.35e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.24e19T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.08e21T + 6.34e41T^{2} \) |
| 53 | \( 1 + 8.38e19T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.82e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.55e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 1.07e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.10e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.47e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 1.49e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.46e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 3.45e23T + 5.42e48T^{2} \) |
| 97 | \( 1 + 3.47e24T + 4.66e49T^{2} \) |
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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
−10.29268359140266962331620846949, −9.597299097331305076700038090025, −8.290673564891981437832614666110, −6.76538880127373206850176936076, −5.56994852653310196315194206249, −5.35454104915113697417631331241, −3.78603625653189207520017236521, −1.75556043678139354200431729313, −1.03546307063396267666116238304, 0,
1.03546307063396267666116238304, 1.75556043678139354200431729313, 3.78603625653189207520017236521, 5.35454104915113697417631331241, 5.56994852653310196315194206249, 6.76538880127373206850176936076, 8.290673564891981437832614666110, 9.597299097331305076700038090025, 10.29268359140266962331620846949
