| L(s) = 1 | + 1.13e4·2-s − 6.34e5·3-s + 9.41e7·4-s − 5.29e8·5-s − 7.16e9·6-s + 6.84e11·8-s − 4.45e11·9-s − 5.98e12·10-s − 1.21e13·11-s − 5.97e13·12-s + 1.33e14·13-s + 3.35e14·15-s + 4.58e15·16-s − 8.48e14·17-s − 5.03e15·18-s + 3.46e15·19-s − 4.98e16·20-s − 1.36e17·22-s − 2.00e16·23-s − 4.34e17·24-s − 1.76e16·25-s + 1.51e18·26-s + 8.19e17·27-s + 6.10e17·29-s + 3.79e18·30-s − 6.65e18·31-s + 2.87e19·32-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 0.688·3-s + 2.80·4-s − 0.969·5-s − 1.34·6-s + 3.52·8-s − 0.525·9-s − 1.89·10-s − 1.16·11-s − 1.93·12-s + 1.59·13-s + 0.668·15-s + 4.06·16-s − 0.353·17-s − 1.02·18-s + 0.359·19-s − 2.72·20-s − 2.27·22-s − 0.190·23-s − 2.42·24-s − 0.0593·25-s + 3.10·26-s + 1.05·27-s + 0.320·29-s + 1.30·30-s − 1.51·31-s + 4.41·32-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
| \(L(13)\) |
\(\approx\) |
\(5.553970764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.553970764\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 - 1.13e4T + 3.35e7T^{2} \) |
| 3 | \( 1 + 6.34e5T + 8.47e11T^{2} \) |
| 5 | \( 1 + 5.29e8T + 2.98e17T^{2} \) |
| 11 | \( 1 + 1.21e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.33e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 8.48e14T + 5.77e30T^{2} \) |
| 19 | \( 1 - 3.46e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 2.00e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 6.10e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 6.65e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 9.20e17T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.01e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 3.45e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 3.45e19T + 6.34e41T^{2} \) |
| 53 | \( 1 - 5.24e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.34e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 2.79e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 9.11e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.20e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 1.71e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 4.35e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 9.54e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.50e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 1.28e25T + 4.66e49T^{2} \) |
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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
−11.16664467541243953717294638795, −10.78326246012170310870934823251, −8.209162308775028894734672840772, −7.13152580324070134283270083583, −5.94133379137316691262674937309, −5.34766768602738736178746133118, −4.16391305205161522695602052435, −3.39873249740142617678765827806, −2.29360261398808203271625194132, −0.74757166990921630234665448751,
0.74757166990921630234665448751, 2.29360261398808203271625194132, 3.39873249740142617678765827806, 4.16391305205161522695602052435, 5.34766768602738736178746133118, 5.94133379137316691262674937309, 7.13152580324070134283270083583, 8.209162308775028894734672840772, 10.78326246012170310870934823251, 11.16664467541243953717294638795
