| L(s) = 1 | + (−0.0346 − 0.169i)2-s + (−0.987 − 0.160i)3-s + (1.81 − 0.772i)4-s + (−2.05 + 0.507i)5-s + (0.00697 + 0.173i)6-s + (−0.935 + 1.97i)7-s + (−0.390 − 0.566i)8-s + (0.948 + 0.316i)9-s + (0.157 + 0.331i)10-s + (−3.67 + 1.22i)11-s + (−1.91 + 0.471i)12-s + (−3.35 − 1.32i)13-s + (0.367 + 0.0904i)14-s + (2.11 − 0.170i)15-s + (2.64 − 2.75i)16-s + (−1.70 + 3.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.0245 − 0.120i)2-s + (−0.569 − 0.0926i)3-s + (0.906 − 0.386i)4-s + (−0.920 + 0.226i)5-s + (0.00284 + 0.0706i)6-s + (−0.353 + 0.745i)7-s + (−0.138 − 0.200i)8-s + (0.316 + 0.105i)9-s + (0.0498 + 0.104i)10-s + (−1.10 + 0.370i)11-s + (−0.552 + 0.136i)12-s + (−0.930 − 0.367i)13-s + (0.0981 + 0.0241i)14-s + (0.545 − 0.0440i)15-s + (0.661 − 0.688i)16-s + (−0.412 + 0.870i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0791611 + 0.243515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0791611 + 0.243515i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.987 + 0.160i)T \) |
| 13 | \( 1 + (3.35 + 1.32i)T \) |
| good | 2 | \( 1 + (0.0346 + 0.169i)T + (-1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (2.05 - 0.507i)T + (4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (0.935 - 1.97i)T + (-4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (3.67 - 1.22i)T + (8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (1.70 - 3.58i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (2.23 + 3.87i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.19 - 3.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.549 - 2.69i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-4.34 - 2.28i)T + (17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (5.58 - 3.53i)T + (15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (9.39 + 1.52i)T + (38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (-3.96 - 2.50i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (0.845 + 6.96i)T + (-45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-2.71 - 3.93i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (2.58 + 2.69i)T + (-2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-2.66 - 0.215i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (9.44 + 4.02i)T + (46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (-6.29 + 7.70i)T + (-14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-0.0827 - 0.0733i)T + (8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (-0.827 - 6.81i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (1.54 - 4.07i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (1.49 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.85 - 6.41i)T + (-81.9 + 51.8i)T^{2} \) |
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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
−11.24083389226392512506497248491, −10.52939302131641281408114518417, −9.833904717116680870805660097938, −8.431494334675284689561328603455, −7.46324386858556650954048845248, −6.76574388290951388210805421723, −5.71824591206850783448754997433, −4.81749718027046046374679042717, −3.21046754691546470233845527541, −2.11596540435291459356277675068,
0.14665941573963436061753670231, 2.43317592240745127604161724015, 3.76376027157889174001206012554, 4.76125579639621102958760052600, 6.04775251830241940159303486284, 7.03378579735779541547161807485, 7.66705629369470058295757520910, 8.472482890646289854007647498539, 10.04653699383478191844916065631, 10.56986139774556776242495345439
