| L(s) = 1 | − 64i·2-s − 729i·3-s − 4.09e3·4-s − 4.66e4·6-s − 1.76e5i·7-s + 2.62e5i·8-s − 5.31e5·9-s + 6.61e6·11-s + 2.98e6i·12-s − 2.40e7i·13-s − 1.12e7·14-s + 1.67e7·16-s + 1.54e8i·17-s + 3.40e7i·18-s − 1.90e8·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.566i·7-s + 0.353i·8-s − 0.333·9-s + 1.12·11-s + 0.288i·12-s − 1.38i·13-s − 0.400·14-s + 0.250·16-s + 1.55i·17-s + 0.235i·18-s − 0.926·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.9774204304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9774204304\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 1.76e5iT - 9.68e10T^{2} \) |
| 11 | \( 1 - 6.61e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.40e7iT - 3.02e14T^{2} \) |
| 17 | \( 1 - 1.54e8iT - 9.90e15T^{2} \) |
| 19 | \( 1 + 1.90e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 3.52e8iT - 5.04e17T^{2} \) |
| 29 | \( 1 - 2.80e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 2.76e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.00e10iT - 2.43e20T^{2} \) |
| 41 | \( 1 + 3.96e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 8.14e10iT - 1.71e21T^{2} \) |
| 47 | \( 1 - 3.41e10iT - 5.46e21T^{2} \) |
| 53 | \( 1 + 2.18e10iT - 2.60e22T^{2} \) |
| 59 | \( 1 + 2.29e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 9.79e9T + 1.61e23T^{2} \) |
| 67 | \( 1 + 7.89e11iT - 5.48e23T^{2} \) |
| 71 | \( 1 + 3.69e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 6.93e11iT - 1.67e24T^{2} \) |
| 79 | \( 1 + 2.23e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.08e12iT - 8.87e24T^{2} \) |
| 89 | \( 1 + 2.22e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.02e13iT - 6.73e25T^{2} \) |
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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
−10.29094413409200704105766556310, −8.820081961937397654481517684382, −8.080722065838660335521094892614, −6.77883050679716528046963030187, −5.78214654279403716618836603771, −4.29443273620275468339408565802, −3.39621564549751829127477002820, −2.08553451741349367033978012506, −1.10259417030089613999697266829, −0.20335093527265095528138856151,
1.34362552205615188246312671796, 2.82701551251812883000837597633, 4.18846023112570730018564796145, 4.91442848764891310970678988864, 6.24925660582494197786296051548, 6.95074247927098264939709913756, 8.444052560233686996015861587067, 9.181872991816047172413051526816, 9.914506668768809167821991530920, 11.48306866605348507961039113809
