| L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.524 + 0.153i)3-s + (−0.654 + 0.755i)4-s + (0.767 + 0.492i)5-s + (−0.357 − 0.412i)6-s + (−0.601 − 4.18i)7-s + (−0.959 − 0.281i)8-s + (−2.27 + 1.46i)9-s + (−0.129 + 0.902i)10-s + (0.630 − 1.38i)11-s + (0.226 − 0.496i)12-s + (−0.0694 + 0.483i)13-s + (3.55 − 2.28i)14-s + (−0.477 − 0.140i)15-s + (−0.142 − 0.989i)16-s + (2.10 + 2.43i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−0.302 + 0.0888i)3-s + (−0.327 + 0.377i)4-s + (0.343 + 0.220i)5-s + (−0.146 − 0.168i)6-s + (−0.227 − 1.58i)7-s + (−0.339 − 0.0996i)8-s + (−0.757 + 0.486i)9-s + (−0.0410 + 0.285i)10-s + (0.190 − 0.416i)11-s + (0.0655 − 0.143i)12-s + (−0.0192 + 0.134i)13-s + (0.950 − 0.611i)14-s + (−0.123 − 0.0362i)15-s + (−0.0355 − 0.247i)16-s + (0.511 + 0.590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.768290 + 0.309467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.768290 + 0.309467i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-2.89 - 3.82i)T \) |
| good | 3 | \( 1 + (0.524 - 0.153i)T + (2.52 - 1.62i)T^{2} \) |
| 5 | \( 1 + (-0.767 - 0.492i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (0.601 + 4.18i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.630 + 1.38i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.0694 - 0.483i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.10 - 2.43i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (3.48 - 4.01i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-5.24 - 6.05i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (7.89 + 2.31i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-6.47 + 4.16i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (7.73 + 4.97i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-4.30 + 1.26i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 0.273T + 47T^{2} \) |
| 53 | \( 1 + (1.04 + 7.26i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.161 + 1.12i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (-13.8 - 4.07i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-0.851 - 1.86i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-4.40 - 9.65i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (0.420 - 0.485i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.230 + 1.60i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (10.0 - 6.47i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (8.97 - 2.63i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (3.83 + 2.46i)T + (40.2 + 88.2i)T^{2} \) |
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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
−16.30649862765667136743546202041, −14.52986563871399418351403026396, −13.94126608632999796814732308481, −12.79670567923278920818246129421, −11.11098853765853939836806092169, −10.11136635555420375502428509324, −8.317447730405059155189793731875, −6.97022123245217403122937559097, −5.65519963003955028082195916220, −3.84906523201256043056889156090,
2.70591599683369626261371552413, 5.12398153294880095300743683635, 6.29454936665876778459962580487, 8.712073639626354723219668266285, 9.588014667734971344146201949001, 11.29041113091205237061651010198, 12.16175684336888974943433514779, 13.04751953556855381136269251696, 14.57762757452994235459042388426, 15.41445287668176296144498779503
