| L(s) = 1 | + (2 + 3.46i)2-s + (−2.92 − 15.3i)3-s + (−7.99 + 13.8i)4-s + (−33.1 + 57.3i)5-s + (47.1 − 40.7i)6-s + (−58.2 − 100. i)7-s − 63.9·8-s + (−225. + 89.7i)9-s − 265.·10-s + (−60.5 − 104. i)11-s + (235. + 81.8i)12-s + (−145. + 251. i)13-s + (233. − 403. i)14-s + (975. + 339. i)15-s + (−128 − 221. i)16-s + 1.53e3·17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.187 − 0.982i)3-s + (−0.249 + 0.433i)4-s + (−0.592 + 1.02i)5-s + (0.535 − 0.462i)6-s + (−0.449 − 0.778i)7-s − 0.353·8-s + (−0.929 + 0.369i)9-s − 0.838·10-s + (−0.150 − 0.261i)11-s + (0.472 + 0.164i)12-s + (−0.238 + 0.412i)13-s + (0.317 − 0.550i)14-s + (1.11 + 0.389i)15-s + (−0.125 − 0.216i)16-s + 1.28·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0290i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.569754612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.569754612\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 - 3.46i)T \) |
| 3 | \( 1 + (2.92 + 15.3i)T \) |
| 11 | \( 1 + (60.5 + 104. i)T \) |
| good | 5 | \( 1 + (33.1 - 57.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (58.2 + 100. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 13 | \( 1 + (145. - 251. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.53e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.20e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-504. + 873. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (3.42e3 + 5.92e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.30e3 - 2.25e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-4.92e3 + 8.52e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-6.60e3 - 1.14e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.15e3 - 2.00e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 2.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-6.50e3 + 1.12e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.31e4 + 2.28e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (4.02e3 - 6.96e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 2.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.36e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-3.66e4 - 6.34e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.96e4 - 3.39e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 5.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.63e4 + 4.56e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−11.70814053671297629723557168215, −10.88937426350596631771183012417, −9.597564614478897488423360770599, −7.944619255709201485037495593149, −7.41384496157845744768487342308, −6.62270434424149070572170500655, −5.58297375918694689514858631579, −3.85920007181684310126768031607, −2.76586617580438363064700391421, −0.66775696895269995562637211967,
0.815493681855524201291098663896, 2.86615357175097734370231615887, 3.94804896617299492465804625790, 5.12501698843459543413016523418, 5.73045816496255164536655961737, 7.77000804243403301895072709110, 9.014285792299645321613701862441, 9.582666739228313604420518785647, 10.64582921490167332299254796294, 11.81720104504332739318424231839
