| L(s) = 1 | + (1.05 + 0.767i)2-s + (0.104 + 0.994i)3-s + (−0.0905 − 0.278i)4-s + (0.841 + 1.45i)5-s + (−0.653 + 1.13i)6-s + (−2.39 − 0.509i)7-s + (0.925 − 2.84i)8-s + (−0.978 + 0.207i)9-s + (−0.229 + 2.18i)10-s + (1.19 − 1.32i)11-s + (0.267 − 0.119i)12-s + (−0.345 − 0.153i)13-s + (−2.14 − 2.38i)14-s + (−1.36 + 0.988i)15-s + (2.69 − 1.95i)16-s + (−0.901 − 1.00i)17-s + ⋯ |
| L(s) = 1 | + (0.747 + 0.543i)2-s + (0.0603 + 0.574i)3-s + (−0.0452 − 0.139i)4-s + (0.376 + 0.651i)5-s + (−0.266 + 0.461i)6-s + (−0.906 − 0.192i)7-s + (0.327 − 1.00i)8-s + (−0.326 + 0.0693i)9-s + (−0.0726 + 0.691i)10-s + (0.360 − 0.400i)11-s + (0.0772 − 0.0344i)12-s + (−0.0957 − 0.0426i)13-s + (−0.572 − 0.636i)14-s + (−0.351 + 0.255i)15-s + (0.673 − 0.489i)16-s + (−0.218 − 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21249 + 0.580274i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21249 + 0.580274i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 31 | \( 1 + (4.95 - 2.54i)T \) |
| good | 2 | \( 1 + (-1.05 - 0.767i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-0.841 - 1.45i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2.39 + 0.509i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-1.19 + 1.32i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.345 + 0.153i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (0.901 + 1.00i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (5.10 - 2.27i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.948 + 2.91i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.60 - 5.52i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (4.70 - 8.15i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.768 - 7.31i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-5.41 + 2.41i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (1.37 - 1.00i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.68 + 0.784i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.39 - 13.2i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 9.06T + 61T^{2} \) |
| 67 | \( 1 + (7.32 + 12.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.65 + 0.351i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-8.60 + 9.55i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (3.92 + 4.35i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.63 + 15.5i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.0964 - 0.296i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.34 + 10.3i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−14.34249444802456914344933905481, −13.46505494188867582198860201560, −12.38775300505546665157406045773, −10.63927909961089402454810146049, −10.08262423490340435860202520064, −8.792880308581572751939988770230, −6.80306637087897885662242112703, −6.16254983813712512543548203057, −4.66714881746597947388822531784, −3.26181823469270347815788660148,
2.31485925125507050997569292313, 3.99106271034785559027189049416, 5.48646086428735932830409969835, 6.88782631116702508598142406407, 8.436986806165265320313919169971, 9.442543324260429480945142603266, 11.00881805508251492758602861558, 12.25091069605091350780067343053, 12.81294636770218404075519662810, 13.46809632492068242458078469319
