| L(s) = 1 | + (0.846 + 0.532i)2-s + (0.919 − 0.577i)3-s + (0.433 + 0.900i)4-s + (−1.69 + 1.46i)5-s + 1.08·6-s + (−2.01 + 2.01i)7-s + (−0.111 + 0.993i)8-s + (−0.790 + 1.64i)9-s + (−2.21 + 0.338i)10-s + (2.48 + 1.19i)11-s + (0.919 + 0.577i)12-s + (3.64 + 0.410i)13-s + (−2.78 + 0.635i)14-s + (−0.709 + 2.32i)15-s + (−0.623 + 0.781i)16-s + (−1.55 + 0.175i)17-s + ⋯ |
| L(s) = 1 | + (0.598 + 0.376i)2-s + (0.530 − 0.333i)3-s + (0.216 + 0.450i)4-s + (−0.756 + 0.654i)5-s + 0.443·6-s + (−0.762 + 0.762i)7-s + (−0.0395 + 0.351i)8-s + (−0.263 + 0.546i)9-s + (−0.698 + 0.107i)10-s + (0.747 + 0.360i)11-s + (0.265 + 0.166i)12-s + (1.01 + 0.113i)13-s + (−0.743 + 0.169i)14-s + (−0.183 + 0.599i)15-s + (−0.155 + 0.195i)16-s + (−0.377 + 0.0425i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0101 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0101 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29529 + 1.30856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29529 + 1.30856i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.846 - 0.532i)T \) |
| 5 | \( 1 + (1.69 - 1.46i)T \) |
| 43 | \( 1 + (6.38 - 1.49i)T \) |
| good | 3 | \( 1 + (-0.919 + 0.577i)T + (1.30 - 2.70i)T^{2} \) |
| 7 | \( 1 + (2.01 - 2.01i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.48 - 1.19i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-3.64 - 0.410i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (1.55 - 0.175i)T + (16.5 - 3.78i)T^{2} \) |
| 19 | \( 1 + (2.12 - 1.02i)T + (11.8 - 14.8i)T^{2} \) |
| 23 | \( 1 + (-6.32 + 2.21i)T + (17.9 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.632 + 2.77i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (0.865 + 3.79i)T + (-27.9 + 13.4i)T^{2} \) |
| 37 | \( 1 + (-7.67 + 7.67i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.207 - 0.908i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (-0.375 - 0.131i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (-0.524 - 4.65i)T + (-51.6 + 11.7i)T^{2} \) |
| 59 | \( 1 + (-4.64 - 3.70i)T + (13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (4.07 + 0.929i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + (-0.834 - 0.291i)T + (52.3 + 41.7i)T^{2} \) |
| 71 | \( 1 + (2.69 + 5.60i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.34 + 11.9i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 - 3.00iT - 79T^{2} \) |
| 83 | \( 1 + (-5.41 - 8.61i)T + (-36.0 + 74.7i)T^{2} \) |
| 89 | \( 1 + (1.56 - 6.83i)T + (-80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + (-4.39 - 12.5i)T + (-75.8 + 60.4i)T^{2} \) |
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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
−11.43247709269065515451096270255, −10.80197615156119256349125058470, −9.287202523524850217988105572485, −8.526204682231360612050544026551, −7.60602382316260602174283327222, −6.66429648068810419856177142114, −5.94260325015039201212309971976, −4.40772690576353348691419888776, −3.35811163979279707346445207422, −2.36590880841929093205152916432,
0.979260103641311603272490398454, 3.27644030119724036600897705109, 3.71947289122340670674585120014, 4.76471873012816856238410047755, 6.21699361315538157775139300840, 7.04611491520614212596736092513, 8.514117811410009172171791383299, 9.041895207022637464723220859487, 10.05305097654995111958923290991, 11.17579774790810281818621012166
