| L(s) = 1 | + (−1.69 + 0.977i)2-s + (1.83 − 0.491i)3-s + (−0.0899 + 0.155i)4-s + (−2.23 − 3.86i)5-s + (−2.62 + 2.62i)6-s + (−4.95 − 4.94i)7-s − 8.16i·8-s + (−4.67 + 2.69i)9-s + (7.55 + 4.36i)10-s + (2.28 + 8.53i)11-s + (−0.0883 + 0.329i)12-s + (18.2 + 18.2i)13-s + (13.2 + 3.53i)14-s + (−5.98 − 5.98i)15-s + (7.62 + 13.2i)16-s + (4.38 + 16.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.846 + 0.488i)2-s + (0.610 − 0.163i)3-s + (−0.0224 + 0.0389i)4-s + (−0.446 − 0.772i)5-s + (−0.437 + 0.437i)6-s + (−0.707 − 0.706i)7-s − 1.02i·8-s + (−0.519 + 0.299i)9-s + (0.755 + 0.436i)10-s + (0.207 + 0.775i)11-s + (−0.00735 + 0.0274i)12-s + (1.40 + 1.40i)13-s + (0.944 + 0.252i)14-s + (−0.399 − 0.399i)15-s + (0.476 + 0.825i)16-s + (0.258 + 0.963i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0913 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0913 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.641431 + 0.585300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.641431 + 0.585300i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (4.95 + 4.94i)T \) |
| 41 | \( 1 + (-5.84 + 40.5i)T \) |
| good | 2 | \( 1 + (1.69 - 0.977i)T + (2 - 3.46i)T^{2} \) |
| 3 | \( 1 + (-1.83 + 0.491i)T + (7.79 - 4.5i)T^{2} \) |
| 5 | \( 1 + (2.23 + 3.86i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.28 - 8.53i)T + (-104. + 60.5i)T^{2} \) |
| 13 | \( 1 + (-18.2 - 18.2i)T + 169iT^{2} \) |
| 17 | \( 1 + (-4.38 - 16.3i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-8.30 - 2.22i)T + (312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (-5.68 - 9.85i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (0.313 - 0.313i)T - 841iT^{2} \) |
| 31 | \( 1 + (-45.2 - 26.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (20.3 + 35.3i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 43 | \( 1 - 66.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (33.8 + 9.08i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-10.7 - 40.2i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (30.4 + 17.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-13.8 - 24.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (0.803 - 0.215i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-29.7 + 29.7i)T - 5.04e3iT^{2} \) |
| 73 | \( 1 + (63.2 - 109. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-39.0 - 10.4i)T + (5.40e3 + 3.12e3i)T^{2} \) |
| 83 | \( 1 + 89.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-21.1 + 79.1i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (48.3 - 48.3i)T - 9.40e3iT^{2} \) |
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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
−11.90678123411371744750200351998, −10.62344457677833558893359211508, −9.506496579469371087531927421729, −8.807594458943003414453252900999, −8.167600131547751585964525587532, −7.21065438915325618942227067840, −6.27383796759424485899678723209, −4.36485805246846643550768922108, −3.50730387246193641466977381271, −1.29464094374092775889196922633,
0.61231807197462363439701327871, 2.86620300138447445569472929543, 3.29442601787509547671724159659, 5.47429204105140401322648397372, 6.40847779285189416145373533427, 8.036613852375797094193619908393, 8.597253437654195887422826098307, 9.454684702467663501034264085217, 10.32867752078392393938964833398, 11.25897083432942243348448597260
