L(s) = 1 | + (3.62 − 2.09i)3-s + (−2.74 − 1.58i)5-s + (−2.24 − 6.63i)7-s + (4.24 − 7.34i)9-s + (−6.62 − 11.4i)11-s + 5.49i·13-s − 13.2·15-s + (−11.7 + 6.77i)17-s + (0.621 + 0.358i)19-s + (−21.9 − 19.3i)21-s + (1.13 − 1.96i)23-s + (−7.48 − 12.9i)25-s + 2.15i·27-s − 20.4·29-s + (21.3 − 12.3i)31-s + ⋯ |
L(s) = 1 | + (1.20 − 0.696i)3-s + (−0.548 − 0.316i)5-s + (−0.320 − 0.947i)7-s + (0.471 − 0.816i)9-s + (−0.601 − 1.04i)11-s + 0.422i·13-s − 0.882·15-s + (−0.690 + 0.398i)17-s + (0.0327 + 0.0188i)19-s + (−1.04 − 0.920i)21-s + (0.0493 − 0.0855i)23-s + (−0.299 − 0.518i)25-s + 0.0797i·27-s − 0.706·29-s + (0.687 − 0.397i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.698331864\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.698331864\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.24 + 6.63i)T \) |
good | 3 | \( 1 + (-3.62 + 2.09i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (2.74 + 1.58i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (6.62 + 11.4i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 5.49iT - 169T^{2} \) |
| 17 | \( 1 + (11.7 - 6.77i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-0.621 - 0.358i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-1.13 + 1.96i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 20.4T + 841T^{2} \) |
| 31 | \( 1 + (-21.3 + 12.3i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-32.4 + 56.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 21.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.48T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-41.3 - 23.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-11.0 - 19.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-72.5 + 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (57.3 + 33.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-46.3 - 80.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-113. + 65.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-38.1 + 66.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (145. + 83.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 25.5iT - 9.40e3T^{2} \) |
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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
−10.59258469257233519612098460894, −9.380472864669899666180128300520, −8.538753242584418817733170829667, −7.85720087905187474464567979434, −7.15229661214220913369188491109, −5.99887961843515260428336146389, −4.33299449951803508645747914633, −3.45495987775247627445337458273, −2.23471666449860843274262726539, −0.58511038728104594873112412585,
2.33750253493311133591020494579, 3.11076421535844472803662783485, 4.24344168368033718221530382200, 5.31670947961665835572101300829, 6.77261337945866564665106813965, 7.83308971422588216265189749430, 8.539283544272564776619581434937, 9.500150553213333548273513028856, 9.993284898723072282001260494810, 11.17141808695311884689802404002