| L(s) = 1 | + (0.124 − 0.708i)2-s + (−0.945 − 0.793i)3-s + (1.39 + 0.507i)4-s + (0.939 − 0.342i)5-s + (−0.680 + 0.570i)6-s + (−0.645 − 1.11i)7-s + (1.25 − 2.16i)8-s + (−0.256 − 1.45i)9-s + (−0.124 − 0.708i)10-s + (−2.88 + 4.99i)11-s + (−0.914 − 1.58i)12-s + (1.80 − 1.51i)13-s + (−0.873 + 0.317i)14-s + (−1.15 − 0.422i)15-s + (0.890 + 0.747i)16-s + (−1.18 + 6.74i)17-s + ⋯ |
| L(s) = 1 | + (0.0883 − 0.501i)2-s + (−0.545 − 0.458i)3-s + (0.696 + 0.253i)4-s + (0.420 − 0.152i)5-s + (−0.277 + 0.233i)6-s + (−0.244 − 0.422i)7-s + (0.442 − 0.767i)8-s + (−0.0854 − 0.484i)9-s + (−0.0395 − 0.224i)10-s + (−0.869 + 1.50i)11-s + (−0.264 − 0.457i)12-s + (0.499 − 0.419i)13-s + (−0.233 + 0.0849i)14-s + (−0.299 − 0.108i)15-s + (0.222 + 0.186i)16-s + (−0.288 + 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.921781 - 0.504715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921781 - 0.504715i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (-2.40 - 3.63i)T \) |
| good | 2 | \( 1 + (-0.124 + 0.708i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (0.945 + 0.793i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (0.645 + 1.11i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.88 - 4.99i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.80 + 1.51i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.18 - 6.74i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (5.32 + 1.93i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.04 + 5.92i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.19 - 2.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + (3.51 + 2.94i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.260 - 0.0947i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.0880 - 0.499i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-6.75 - 2.45i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.00 + 5.67i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.77 - 2.46i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.77 + 10.0i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.05 + 0.382i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.83 - 1.53i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (10.7 + 9.05i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.608 + 1.05i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.85 - 5.74i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.83 - 10.4i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−13.35796093485149194523698591361, −12.56348189825443716865511018563, −11.98647917142048389803120160915, −10.54391173781401413019627030593, −9.996736311961358065817189739657, −8.046360171068981311960587157628, −6.87266649419570546830203349664, −5.84634900659811758936370419757, −3.85032296757522803202775280847, −1.88049282252481349018609039789,
2.72481773683932983646792641885, 5.20352106939321164766957813731, 5.85406021095989433924118443708, 7.16584893839632514019376523080, 8.581357206963133559745026937096, 10.03171586140055639764490152987, 11.11617112555781409245304413787, 11.59720803876963175851446683009, 13.52050477430700937285920327581, 14.04089761568424539215082271661
