| L(s) = 1 | + (0.173 + 0.984i)2-s + (0.0338 + 1.73i)3-s + (−0.939 + 0.342i)4-s + (0.0666 − 0.183i)5-s + (−1.69 + 0.334i)6-s + (1.64 + 2.07i)7-s + (−0.5 − 0.866i)8-s + (−2.99 + 0.117i)9-s + (0.192 + 0.0338i)10-s + (−4.42 − 2.55i)11-s + (−0.624 − 1.61i)12-s + (−1.65 − 4.55i)13-s + (−1.75 + 1.98i)14-s + (0.319 + 0.109i)15-s + (0.766 − 0.642i)16-s + (−2.40 + 6.61i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.0195 + 0.999i)3-s + (−0.469 + 0.171i)4-s + (0.0298 − 0.0819i)5-s + (−0.693 + 0.136i)6-s + (0.621 + 0.783i)7-s + (−0.176 − 0.306i)8-s + (−0.999 + 0.0390i)9-s + (0.0607 + 0.0107i)10-s + (−1.33 − 0.771i)11-s + (−0.180 − 0.466i)12-s + (−0.459 − 1.26i)13-s + (−0.468 + 0.529i)14-s + (0.0825 + 0.0282i)15-s + (0.191 − 0.160i)16-s + (−0.584 + 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.279095 - 0.614187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.279095 - 0.614187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.0338 - 1.73i)T \) |
| 7 | \( 1 + (-1.64 - 2.07i)T \) |
| 19 | \( 1 + (4.34 + 0.326i)T \) |
| good | 5 | \( 1 + (-0.0666 + 0.183i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (4.42 + 2.55i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.65 + 4.55i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.40 - 6.61i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (5.56 - 6.63i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.51 + 1.27i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + 0.481iT - 31T^{2} \) |
| 37 | \( 1 + (-0.660 - 0.381i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.40 - 3.06i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.13 + 6.44i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.279 - 0.766i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.36 - 0.861i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (6.01 + 2.18i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.82 - 6.56i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.32 + 0.233i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.09 - 11.8i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.527 + 2.99i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.73 - 3.25i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (5.01 + 2.89i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.163 - 0.928i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.63 + 3.13i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.69445232260397144534809808054, −9.994680571110706667645613312809, −8.867512604905423493867448759021, −8.276238575748876990527020909026, −7.72113334533733581788771808075, −5.91249126818594005597696633174, −5.65793935011700476809792955350, −4.73703478358964703381024136063, −3.62937758684881036821024212597, −2.45298388427867751922184685875,
0.29734307568944160100871304738, 2.01868776296321869301353789485, 2.59198934043615569977803268467, 4.38663160480418260815788637752, 4.91863585900410400515201670289, 6.40100295913360138833850678033, 7.19585369585066046265804447046, 7.946394972687725215464267868640, 8.855178943603834412041758140873, 9.890611540101993929231179307406
