| L(s) = 1 | + (1.08 − 2.63i)2-s + (2.88 − 4.32i)3-s + (−2.90 − 2.90i)4-s + (0.711 + 3.57i)5-s + (−8.22 − 12.3i)6-s + (0.644 − 3.23i)7-s + (−0.288 + 0.119i)8-s + (−6.88 − 16.6i)9-s + (10.1 + 2.02i)10-s + (−2.02 + 1.35i)11-s + (−20.9 + 4.16i)12-s + (−7.73 + 7.73i)13-s + (−7.81 − 5.22i)14-s + (17.5 + 7.24i)15-s − 15.5i·16-s + ⋯ |
| L(s) = 1 | + (0.544 − 1.31i)2-s + (0.962 − 1.44i)3-s + (−0.726 − 0.726i)4-s + (0.142 + 0.715i)5-s + (−1.37 − 2.05i)6-s + (0.0920 − 0.462i)7-s + (−0.0360 + 0.0149i)8-s + (−0.765 − 1.84i)9-s + (1.01 + 0.202i)10-s + (−0.183 + 0.122i)11-s + (−1.74 + 0.347i)12-s + (−0.594 + 0.594i)13-s + (−0.558 − 0.373i)14-s + (1.16 + 0.483i)15-s − 0.971i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.222570 - 2.91901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.222570 - 2.91901i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (-1.08 + 2.63i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-2.88 + 4.32i)T + (-3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (-0.711 - 3.57i)T + (-23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (-0.644 + 3.23i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (2.02 - 1.35i)T + (46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (7.73 - 7.73i)T - 169iT^{2} \) |
| 19 | \( 1 + (3.50 - 8.45i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-1.91 - 2.87i)T + (-202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-23.5 + 4.69i)T + (776. - 321. i)T^{2} \) |
| 31 | \( 1 + (-23.0 - 15.3i)T + (367. + 887. i)T^{2} \) |
| 37 | \( 1 + (-21.3 + 32.0i)T + (-523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (15.0 - 75.4i)T + (-1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (-20.7 - 50.0i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (5.13 - 5.13i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-21.8 + 52.6i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (21.2 - 8.82i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (40.6 + 8.09i)T + (3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 + 17.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-28.6 + 42.8i)T + (-1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (12.1 + 60.9i)T + (-4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (98.5 - 65.8i)T + (2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (86.9 + 36.0i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-19.6 - 19.6i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (14.9 - 2.97i)T + (8.69e3 - 3.60e3i)T^{2} \) |
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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
−11.40276948420414108725803743056, −10.38845790419811951320867279350, −9.458057415706135464823302755373, −8.115240525676659043963022290261, −7.25485356030258190512869936189, −6.41604776141427753270601767843, −4.48247191691230275999023929267, −3.11343017794663586721160867850, −2.40705113826629068923457604668, −1.24170247012536108387269679254,
2.70035105869044123455695607582, 4.18937300737898038025429596129, 4.96404713547455746782859143481, 5.68457833607845876332825975930, 7.25672109885631172057706450198, 8.440682002332583579125439938973, 8.775324210460936643606685900901, 9.927936587599135838064689298343, 10.78111124498667727185841489089, 12.31357331561368329693674189320
