| L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−1.73 − i)5-s − 2.82i·8-s + (1.41 + 2.44i)10-s + (−2.44 + 1.41i)11-s + 12.7·13-s + (−2.00 + 3.46i)16-s + (−19.0 + 11i)17-s + (2.82 − 4.89i)19-s − 3.99i·20-s + 4·22-s + (−2.44 − 1.41i)23-s + (−10.5 − 18.1i)25-s + (−15.5 − 9i)26-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.346 − 0.200i)5-s − 0.353i·8-s + (0.141 + 0.244i)10-s + (−0.222 + 0.128i)11-s + 0.979·13-s + (−0.125 + 0.216i)16-s + (−1.12 + 0.647i)17-s + (0.148 − 0.257i)19-s − 0.199i·20-s + 0.181·22-s + (−0.106 − 0.0614i)23-s + (−0.419 − 0.727i)25-s + (−0.599 − 0.346i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3466190068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3466190068\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (1.73 + i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (2.44 - 1.41i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 12.7T + 169T^{2} \) |
| 17 | \( 1 + (19.0 - 11i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-2.82 + 4.89i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (2.44 + 1.41i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 35.3iT - 841T^{2} \) |
| 31 | \( 1 + (-16.9 - 29.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (32 - 55.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 20iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 44T + 1.84e3T^{2} \) |
| 47 | \( 1 + (58.8 + 34i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-15.9 + 9.19i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (86.6 - 50i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (26.1 - 45.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (60 + 103. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 8.48iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-37.4 - 64.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (46 - 79.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 112iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-17.3 - 10i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 26.8T + 9.40e3T^{2} \) |
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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
−10.28797475138409649124636065037, −9.361319760282265887204826853353, −8.432446524318259723657515103213, −8.073845308482731796128299332117, −6.82765060082547342259978732061, −6.11401332225561430700175720988, −4.70429616913885094185580186868, −3.82567465059602581224905752358, −2.61540403785590266147100581994, −1.33896819833987933303216874678,
0.14461982531044002103447218437, 1.69264840740555270370624482902, 3.09099328351177024526075893737, 4.24746524683604579624667606311, 5.44485204326174577687466063000, 6.32402301133214300290007425235, 7.22462485879531479606872749140, 7.926332980712882221344083322021, 8.897399638810997083677982376077, 9.399117807491630322193410049961
