| L(s) = 1 | + (−0.730 − 1.86i)2-s + (0.672 + 8.97i)3-s + (−2.93 + 2.72i)4-s + (−11.3 + 7.76i)5-s + (16.2 − 7.81i)6-s + (9.97 − 15.6i)7-s + (7.20 + 3.47i)8-s + (−53.4 + 8.05i)9-s + (22.7 + 15.5i)10-s + (−57.9 − 8.73i)11-s + (−26.4 − 24.4i)12-s + (40.2 − 50.4i)13-s + (−36.3 − 7.17i)14-s + (−77.3 − 96.9i)15-s + (1.19 − 15.9i)16-s + (−10.8 + 3.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.258 − 0.658i)2-s + (0.129 + 1.72i)3-s + (−0.366 + 0.340i)4-s + (−1.01 + 0.694i)5-s + (1.10 − 0.531i)6-s + (0.538 − 0.842i)7-s + (0.318 + 0.153i)8-s + (−1.98 + 0.298i)9-s + (0.719 + 0.490i)10-s + (−1.58 − 0.239i)11-s + (−0.635 − 0.589i)12-s + (0.858 − 1.07i)13-s + (−0.693 − 0.136i)14-s + (−1.33 − 1.66i)15-s + (0.0186 − 0.249i)16-s + (−0.155 + 0.0478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0691i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0152975 + 0.441836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0152975 + 0.441836i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.730 + 1.86i)T \) |
| 7 | \( 1 + (-9.97 + 15.6i)T \) |
| good | 3 | \( 1 + (-0.672 - 8.97i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (11.3 - 7.76i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (57.9 + 8.73i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-40.2 + 50.4i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (10.8 - 3.35i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (72.7 - 126. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-8.09 - 2.49i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (60.9 - 266. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (10.6 + 18.5i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (118. + 110. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (8.69 + 4.18i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (82.0 - 39.5i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-99.7 - 254. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (52.4 - 48.6i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (438. + 298. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-115. - 107. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-440. - 762. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (8.32 + 36.4i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-89.8 + 228. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (313. - 542. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-872. - 1.09e3i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-37.4 + 5.64i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + 525.T + 9.12e5T^{2} \) |
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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
−14.18541109075689578271052615756, −12.76935821654229138921573371949, −11.05215469889799860556795666744, −10.77588054416957160414849682469, −10.17291193045758862159976441980, −8.491204489748092774172128689970, −7.77547190891377506835122275931, −5.30753110640650110345340093759, −3.96029345766811660293951910710, −3.22057289407754908043737106109,
0.26298192583821180163483837849, 2.15190609023816951085431677661, 4.81410758201293631065736799809, 6.25134885057379776809039499640, 7.44843773206935574310603969201, 8.256930464412953985799045360115, 8.856420881045409781384125136659, 11.18543446683171259649631028475, 12.03281787336983496513036937334, 13.04894699704907121440990432179
