| L(s) = 1 | + 64i·2-s − 729i·3-s − 4.09e3·4-s + 4.66e4·6-s + 2.77e5i·7-s − 2.62e5i·8-s − 5.31e5·9-s + 1.89e4·11-s + 2.98e6i·12-s + 6.01e6i·13-s − 1.77e7·14-s + 1.67e7·16-s + 9.37e7i·17-s − 3.40e7i·18-s + 5.41e7·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.889i·7-s − 0.353i·8-s − 0.333·9-s + 0.00322·11-s + 0.288i·12-s + 0.345i·13-s − 0.629·14-s + 0.250·16-s + 0.942i·17-s − 0.235i·18-s + 0.263·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.6351356517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6351356517\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 64iT \) |
| 3 | \( 1 + 729iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.77e5iT - 9.68e10T^{2} \) |
| 11 | \( 1 - 1.89e4T + 3.45e13T^{2} \) |
| 13 | \( 1 - 6.01e6iT - 3.02e14T^{2} \) |
| 17 | \( 1 - 9.37e7iT - 9.90e15T^{2} \) |
| 19 | \( 1 - 5.41e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 2.03e8iT - 5.04e17T^{2} \) |
| 29 | \( 1 + 3.05e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 4.34e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 9.81e9iT - 2.43e20T^{2} \) |
| 41 | \( 1 + 7.81e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.58e10iT - 1.71e21T^{2} \) |
| 47 | \( 1 + 4.89e9iT - 5.46e21T^{2} \) |
| 53 | \( 1 - 4.29e10iT - 2.60e22T^{2} \) |
| 59 | \( 1 - 2.79e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 2.71e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 2.60e11iT - 5.48e23T^{2} \) |
| 71 | \( 1 - 1.75e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.00e12iT - 1.67e24T^{2} \) |
| 79 | \( 1 + 3.96e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 6.14e11iT - 8.87e24T^{2} \) |
| 89 | \( 1 + 4.36e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 4.72e12iT - 6.73e25T^{2} \) |
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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
−11.36194709614394191361190262781, −9.956238752382344641463706283859, −8.864205226986728959395979516557, −8.124310068596188656978001227759, −7.01477402418503796553537338074, −6.06833111686468750775859277843, −5.23077841594208087259279703428, −3.82372430207192990986325736132, −2.43573266295297670712789182765, −1.28466757967830769187509286947,
0.13624238182852039020254171291, 1.06085486972939136081634916534, 2.51254088795644306126035277897, 3.56844490579942271298234099676, 4.48012748589652471026949115881, 5.51109124589768791551403144667, 7.01778825720110275462934851368, 8.149119159594918272725129141593, 9.347279169154615925422463447794, 10.10561291241388335298676708838
