| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.324 + 1.70i)3-s + (0.173 − 0.984i)4-s + (2.22 − 0.392i)5-s + (0.845 + 1.51i)6-s + (−1.16 + 2.02i)7-s + (−0.500 − 0.866i)8-s + (−2.78 − 1.10i)9-s + (1.45 − 1.73i)10-s + (2.52 − 1.45i)11-s + (1.61 + 0.614i)12-s + (−0.451 − 1.24i)13-s + (0.405 + 2.30i)14-s + (−0.0536 + 3.91i)15-s + (−0.939 − 0.342i)16-s + (−3.72 − 4.43i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.187 + 0.982i)3-s + (0.0868 − 0.492i)4-s + (0.996 − 0.175i)5-s + (0.345 + 0.617i)6-s + (−0.441 + 0.764i)7-s + (−0.176 − 0.306i)8-s + (−0.929 − 0.367i)9-s + (0.459 − 0.548i)10-s + (0.760 − 0.438i)11-s + (0.467 + 0.177i)12-s + (−0.125 − 0.344i)13-s + (0.108 + 0.614i)14-s + (−0.0138 + 1.01i)15-s + (−0.234 − 0.0855i)16-s + (−0.902 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36096 + 0.0572460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36096 + 0.0572460i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.324 - 1.70i)T \) |
| 19 | \( 1 + (1.79 - 3.97i)T \) |
| good | 5 | \( 1 + (-2.22 + 0.392i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (1.16 - 2.02i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.52 + 1.45i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.451 + 1.24i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (3.72 + 4.43i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (8.06 + 1.42i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.64 - 1.38i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-5.27 - 3.04i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.98iT - 37T^{2} \) |
| 41 | \( 1 + (-8.52 - 3.10i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0666 + 0.377i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.57 + 7.83i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.494 - 2.80i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (2.53 - 2.12i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.01 + 5.77i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (10.4 - 12.4i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.29 - 13.0i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.84 - 2.12i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.77 + 4.87i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (1.62 + 0.938i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.82 - 2.11i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.13 - 7.30i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−13.78260200942921795021670780812, −12.43455694334962448189311742300, −11.61227965452281819115887350365, −10.36534213437305057347560661064, −9.586090623844635489213349599868, −8.722907441691363473963001816075, −6.26057158212683384108046306341, −5.58868368907609430127246149325, −4.17023054091789406898394259438, −2.57545820937354260546543606300,
2.16018919362293811871625935937, 4.26328544805486197974466763746, 6.13939144808346938815779246750, 6.53704944942956470006518715219, 7.78204152463622150936314832165, 9.214469885153326827761371495409, 10.53050456132241953562572649302, 11.81766463678919820198861039329, 12.83776482657488887597324097504, 13.67590985449466399352771536787
