| L(s) = 1 | + (−0.666 + 0.983i)2-s + (−1.73 + 0.0740i)3-s + (0.217 + 0.547i)4-s + (1.67 − 3.62i)5-s + (1.08 − 1.75i)6-s + (−0.654 − 2.35i)7-s + (−3.00 − 0.661i)8-s + (2.98 − 0.256i)9-s + (2.44 + 4.06i)10-s + (2.14 + 2.03i)11-s + (−0.417 − 0.930i)12-s + (0.353 − 3.25i)13-s + (2.75 + 0.927i)14-s + (−2.63 + 6.39i)15-s + (1.79 − 1.70i)16-s + (7.44 + 2.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.471 + 0.695i)2-s + (−0.999 + 0.0427i)3-s + (0.108 + 0.273i)4-s + (0.749 − 1.61i)5-s + (0.441 − 0.714i)6-s + (−0.247 − 0.890i)7-s + (−1.06 − 0.233i)8-s + (0.996 − 0.0854i)9-s + (0.772 + 1.28i)10-s + (0.646 + 0.612i)11-s + (−0.120 − 0.268i)12-s + (0.0981 − 0.902i)13-s + (0.735 + 0.247i)14-s + (−0.679 + 1.65i)15-s + (0.449 − 0.425i)16-s + (1.80 + 0.501i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.761255 - 0.0913010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.761255 - 0.0913010i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.73 - 0.0740i)T \) |
| 59 | \( 1 + (-3.28 - 6.94i)T \) |
| good | 2 | \( 1 + (0.666 - 0.983i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (-1.67 + 3.62i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (0.654 + 2.35i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (-2.14 - 2.03i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.353 + 3.25i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (-7.44 - 2.06i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (0.902 + 0.685i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (2.77 + 5.24i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (2.57 - 1.74i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-1.31 - 1.72i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-0.660 - 3.00i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (0.389 + 0.206i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (-1.20 - 1.27i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-2.64 + 1.22i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (-0.957 + 1.59i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (7.14 + 4.84i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (2.69 - 12.2i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (3.79 + 8.20i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (4.53 - 13.4i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (0.288 - 5.32i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (0.561 - 3.42i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (-3.97 - 5.85i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (0.0324 + 0.0963i)T + (-77.2 + 58.7i)T^{2} \) |
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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
−12.48802622967602933856730469824, −12.08335590915205962713871522056, −10.33249821094873347170924698732, −9.683643706841200607390650612389, −8.495056097857826468323712231757, −7.46922493876791555291627627889, −6.27351151459249164574332352987, −5.38489906300143036682645146276, −4.06349358457386604430667616134, −1.01967648550933118412722720473,
1.84212578573219306999561040657, 3.29465884084875084790488000596, 5.87940858823124766720604042056, 5.98277411659593363343182257295, 7.28560707892353576911962467148, 9.341596482336440821234186109184, 9.889121041445127652437564603676, 10.80607011915542323853201437926, 11.60276431400872011878642888572, 12.10680819861637467152154427316
