L(s) = 1 | − 11.6·2-s + 15.5i·3-s + 70.9·4-s − 190. i·5-s − 181. i·6-s − 80.4·8-s − 243·9-s + 2.21e3i·10-s − 2.05e3·11-s + 1.10e3i·12-s + 3.05e3i·13-s + 2.97e3·15-s − 3.60e3·16-s − 2.85e3i·17-s + 2.82e3·18-s − 3.94e3i·19-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 0.577i·3-s + 1.10·4-s − 1.52i·5-s − 0.838i·6-s − 0.157·8-s − 0.333·9-s + 2.21i·10-s − 1.54·11-s + 0.639i·12-s + 1.39i·13-s + 0.881·15-s − 0.879·16-s − 0.580i·17-s + 0.483·18-s − 0.575i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4808458485\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4808458485\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 15.5iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 11.6T + 64T^{2} \) |
| 5 | \( 1 + 190. iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 2.05e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 3.05e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.85e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.94e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 660.T + 1.48e8T^{2} \) |
| 29 | \( 1 + 9.28e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.80e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.69e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 6.79e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.23e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.47e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.19e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.92e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.32e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 3.49e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.05e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.36e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 5.86e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.06e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.30e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.05e5iT - 8.32e11T^{2} \) |
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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
−11.74952367677392697488588098798, −10.75827301695586625939022674734, −9.684875960524517336173884969316, −9.024847445711395712707358853816, −8.314090936667349930962434655282, −7.19916782457905686897216874884, −5.32494517350602127565005684863, −4.41274518773843202489639465003, −2.19276037064304946665000035523, −0.73408990079405957977637201776,
0.35608495680581234003829326691, 2.08356913050186560405149184027, 3.13989877508308713327576153302, 5.62299572727144211423161531249, 6.91402208462778974015499238130, 7.73234855061136440431400401269, 8.363452762014134459974183097385, 10.08985798266490371048751319394, 10.45402322318675725541209075501, 11.27462504042294674074909083388