L(s) = 1 | + (−1.27 + 2.20i)2-s + (−0.937 − 1.62i)3-s + (−2.25 − 3.90i)4-s + (−0.5 + 0.866i)5-s + 4.78·6-s + (−2.44 − 1.01i)7-s + 6.38·8-s + (−0.257 + 0.445i)9-s + (−1.27 − 2.20i)10-s + (0.5 + 0.866i)11-s + (−4.22 + 7.31i)12-s + 1.50·13-s + (5.35 − 4.10i)14-s + 1.87·15-s + (−3.63 + 6.30i)16-s + (2.79 + 4.83i)17-s + ⋯ |
L(s) = 1 | + (−0.901 + 1.56i)2-s + (−0.541 − 0.937i)3-s + (−1.12 − 1.95i)4-s + (−0.223 + 0.387i)5-s + 1.95·6-s + (−0.923 − 0.383i)7-s + 2.25·8-s + (−0.0856 + 0.148i)9-s + (−0.403 − 0.698i)10-s + (0.150 + 0.261i)11-s + (−1.21 + 2.11i)12-s + 0.417·13-s + (1.43 − 1.09i)14-s + 0.484·15-s + (−0.909 + 1.57i)16-s + (0.677 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.178692 + 0.393024i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.178692 + 0.393024i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.44 + 1.01i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (1.27 - 2.20i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.937 + 1.62i)T + (-1.5 + 2.59i)T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + (-2.79 - 4.83i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.272 - 0.472i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.29 - 7.43i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + (-1.92 - 3.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.95 - 6.84i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 + (-3.49 + 6.05i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.34 + 4.06i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.60 - 7.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.70 + 6.41i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.97 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + (-3.62 - 6.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.11 - 8.86i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.53T + 83T^{2} \) |
| 89 | \( 1 + (-0.781 + 1.35i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.93T + 97T^{2} \) |
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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
−11.67071408997759629314231583766, −10.30048155429136661577934184107, −9.788538323075918410593829089211, −8.564269846985952110544050261065, −7.66086703133671548407037922106, −6.97096123574632027388802458542, −6.28667189051155742186246684939, −5.63566025080408865319557714974, −3.81045916165810580142045124682, −1.21434764310688115548718368411,
0.48779011974449063858423807209, 2.53097502832665477458911908545, 3.69839579489218465084139856085, 4.62415047058675571620082675479, 6.01294691153688377355956223740, 7.69515762066842434351769145952, 8.817345933723124941874606096814, 9.441314414164836569504855709915, 10.18098799595011299404143389526, 10.82848570920168430021636733965