L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s + (−2.5 + 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + (−0.499 − 0.866i)12-s + 13-s + (2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.408·6-s + (−0.944 + 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + (−0.144 − 0.249i)12-s + 0.277·13-s + (0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + (−0.117 + 0.204i)18-s + (−0.114 − 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.020343263\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020343263\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 17T + 97T^{2} \) |
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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
−9.878539762188459890328668583093, −9.070687482885223001763757922645, −8.697693142118292960419602707999, −7.45260312017594007889447383619, −6.29677017624464986440904984779, −5.74316879069052146843386764901, −4.34333851147284749707892233547, −3.53535842091399441853401678485, −2.60569168306205982818148576472, −0.838615799817317477900438722436,
0.835910673877510865392407794410, 2.35720735954868266836583136696, 3.85616744471844998033939457276, 4.85646338391146831939142167997, 6.00990740487378164892421728331, 6.70433383303470846135701217114, 7.22686566641299084553236629991, 8.172840002616628145709527841324, 9.186146840794650617902148599649, 9.762356199163517965747611310847