L(s) = 1 | − 2.60e3·2-s − 4.93e5·3-s − 2.67e7·4-s + 3.09e8·5-s + 1.28e9·6-s + 1.57e11·8-s − 6.03e11·9-s − 8.05e11·10-s + 9.78e12·11-s + 1.32e13·12-s − 1.58e14·13-s − 1.52e14·15-s + 4.89e14·16-s + 2.90e15·17-s + 1.57e15·18-s − 1.67e16·19-s − 8.28e15·20-s − 2.54e16·22-s + 1.60e17·23-s − 7.75e16·24-s − 2.02e17·25-s + 4.12e17·26-s + 7.16e17·27-s + 1.12e18·29-s + 3.97e17·30-s − 1.46e17·31-s − 6.54e18·32-s + ⋯ |
L(s) = 1 | − 0.449·2-s − 0.536·3-s − 0.797·4-s + 0.566·5-s + 0.241·6-s + 0.808·8-s − 0.712·9-s − 0.254·10-s + 0.939·11-s + 0.428·12-s − 1.88·13-s − 0.304·15-s + 0.434·16-s + 1.21·17-s + 0.320·18-s − 1.73·19-s − 0.452·20-s − 0.422·22-s + 1.52·23-s − 0.433·24-s − 0.678·25-s + 0.847·26-s + 0.918·27-s + 0.588·29-s + 0.136·30-s − 0.0333·31-s − 1.00·32-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
\(L(13)\) |
\(\approx\) |
\(0.6704707929\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6704707929\) |
\(L(\frac{27}{2})\) |
|
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 | \( 1 + 2.60e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 4.93e5T + 8.47e11T^{2} \) |
| 5 | \( 1 - 3.09e8T + 2.98e17T^{2} \) |
| 11 | \( 1 - 9.78e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.58e14T + 7.05e27T^{2} \) |
| 17 | \( 1 - 2.90e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.67e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.60e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.12e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 1.46e17T + 1.92e37T^{2} \) |
| 37 | \( 1 + 2.78e18T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.26e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.58e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 4.78e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 4.38e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.71e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 1.89e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 6.63e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 2.36e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.17e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 7.85e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 4.07e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 3.03e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 7.83e24T + 4.66e49T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.67677153431658425652090406983, −9.708530644251513412014855882897, −8.926729413295858086945366972485, −7.69960999939978282222924461714, −6.37868636081653304185015914500, −5.26480935468962085650179268148, −4.42200399847170676809115214891, −2.86741831962622919659657252060, −1.52195421540971887158138478698, −0.39699950675252461574221265553,
0.39699950675252461574221265553, 1.52195421540971887158138478698, 2.86741831962622919659657252060, 4.42200399847170676809115214891, 5.26480935468962085650179268148, 6.37868636081653304185015914500, 7.69960999939978282222924461714, 8.926729413295858086945366972485, 9.708530644251513412014855882897, 10.67677153431658425652090406983