| L(s) = 1 | + (−0.892 + 1.54i)2-s + (0.172 − 0.978i)3-s + (−0.591 − 1.02i)4-s + (3.13 − 1.14i)5-s + (1.35 + 1.13i)6-s + (−0.0641 + 0.0233i)7-s − 1.45·8-s + (1.89 + 0.688i)9-s + (−1.03 + 5.86i)10-s + (0.356 + 2.02i)11-s + (−1.10 + 0.402i)12-s + (−4.11 + 1.49i)13-s + (0.0211 − 0.119i)14-s + (−0.576 − 3.26i)15-s + (2.48 − 4.30i)16-s + (2.90 − 5.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 + 1.09i)2-s + (0.0996 − 0.565i)3-s + (−0.295 − 0.512i)4-s + (1.40 − 0.510i)5-s + (0.554 + 0.465i)6-s + (−0.0242 + 0.00882i)7-s − 0.515·8-s + (0.630 + 0.229i)9-s + (−0.327 + 1.85i)10-s + (0.107 + 0.609i)11-s + (−0.318 + 0.116i)12-s + (−1.14 + 0.415i)13-s + (0.00565 − 0.0320i)14-s + (−0.148 − 0.843i)15-s + (0.620 − 1.07i)16-s + (0.705 − 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.847469 + 0.397558i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.847469 + 0.397558i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-10.4 - 0.0442i)T \) |
| good | 2 | \( 1 + (0.892 - 1.54i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.172 + 0.978i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (-3.13 + 1.14i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.0641 - 0.0233i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.356 - 2.02i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (4.11 - 1.49i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.90 + 5.03i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.55 - 6.15i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.28 + 3.95i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.288 + 1.63i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (5.17 + 1.88i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (2.41 + 0.878i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + 4.80T + 41T^{2} \) |
| 43 | \( 1 + (2.63 - 4.55i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.00 - 4.20i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-4.68 + 1.70i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.57 + 8.94i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.21 - 2.70i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-11.8 - 4.29i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.16 - 5.47i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.13 + 12.0i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.28 - 0.832i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.495 + 2.81i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-5.81 - 4.87i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (15.4 - 5.62i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.09888178835584047675601937494, −12.80515698446298546449660079994, −12.16748114185667606907214006617, −9.933042703379440755499850792238, −9.593176944686325237674281023957, −8.190673556042008379203181460884, −7.18636234301652081564053153370, −6.24351097234271445478697521353, −4.99145095655395643546339996424, −2.04707784574946067947724445885,
1.93935966187841424466265119017, 3.38506900493214826059143020200, 5.43474882557164321321931289375, 6.75271182273526567754046844403, 8.702821526778762764049119947869, 9.756010147312328323140881391139, 10.18690300583403706288776780306, 11.01148022816296425465448356991, 12.40355013036941829837576080763, 13.31982359118544671884019889173
