L(s) = 1 | + (−1.5 − 0.866i)3-s + (1 + 1.73i)4-s + (−2 + 1.73i)7-s + (1.5 + 2.59i)9-s − 3.46i·12-s + (−2.5 + 2.59i)13-s + (−1.99 + 3.46i)16-s + (4 + 6.92i)19-s + (4.5 − 0.866i)21-s + 5·25-s − 5.19i·27-s + (−5 − 1.73i)28-s − 7·31-s + (−3 + 5.19i)36-s + (6 + 3.46i)37-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (0.5 + 0.866i)4-s + (−0.755 + 0.654i)7-s + (0.5 + 0.866i)9-s − 0.999i·12-s + (−0.693 + 0.720i)13-s + (−0.499 + 0.866i)16-s + (0.917 + 1.58i)19-s + (0.981 − 0.188i)21-s + 25-s − 0.999i·27-s + (−0.944 − 0.327i)28-s − 1.25·31-s + (−0.5 + 0.866i)36-s + (0.986 + 0.569i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0396 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0396 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.613041 + 0.589225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.613041 + 0.589225i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
| 13 | \( 1 + (2.5 - 2.59i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + (-6 - 3.46i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.5 + 11.2i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.5 + 4.33i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.5 + 6.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 17T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.18268868208400242859850209731, −11.52425857137614495837828872205, −10.40089889788438293275257400370, −9.300189100969128268134180732686, −8.034897633269054325610519937778, −7.10419754443373669064748729414, −6.32244065960848665532550068497, −5.19047948539486455925557590928, −3.57592246721609012130707019413, −2.09902232725384002936870075095,
0.71402027238115935797487943558, 3.02829969689145363963164564532, 4.67737010049739350255090035996, 5.55437350422619564645922677295, 6.65428466978253725762091186149, 7.33232074103355178450606162694, 9.328453358891709060506405346161, 9.870708903496518587277158065707, 10.80539758213148015010683110750, 11.33803261408755340200972346292