L(s) = 1 | + 3.53i·3-s − 7.77·7-s − 3.49·9-s − 10.7i·11-s + 10.7i·13-s + 21.5i·17-s + 18.1·19-s − 27.4i·21-s − 13.0i·23-s + 19.4i·27-s − 38.9i·29-s + (26.0 − 16.7i)31-s + 38.0·33-s − 48.3i·37-s − 37.8·39-s + ⋯ |
L(s) = 1 | + 1.17i·3-s − 1.11·7-s − 0.387·9-s − 0.979i·11-s + 0.824i·13-s + 1.26i·17-s + 0.954·19-s − 1.30i·21-s − 0.568i·23-s + 0.721i·27-s − 1.34i·29-s + (0.841 − 0.540i)31-s + 1.15·33-s − 1.30i·37-s − 0.970·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.608291812\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.608291812\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + (-26.0 + 16.7i)T \) |
good | 3 | \( 1 - 3.53iT - 9T^{2} \) |
| 7 | \( 1 + 7.77T + 49T^{2} \) |
| 11 | \( 1 + 10.7iT - 121T^{2} \) |
| 13 | \( 1 - 10.7iT - 169T^{2} \) |
| 17 | \( 1 - 21.5iT - 289T^{2} \) |
| 19 | \( 1 - 18.1T + 361T^{2} \) |
| 23 | \( 1 + 13.0iT - 529T^{2} \) |
| 29 | \( 1 + 38.9iT - 841T^{2} \) |
| 37 | \( 1 + 48.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 63.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 2.27iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 72.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 117.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 20.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 15.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 61.1T + 5.04e3T^{2} \) |
| 73 | \( 1 - 40.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 95.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 28.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 16.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 171.T + 9.40e3T^{2} \) |
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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
−8.689865635277826544772947381272, −8.100661170429066588805980078954, −6.87849445965970561212806545707, −6.28462280476993386359322928345, −5.52958105607650719180521460618, −4.60103855212605670437600994934, −3.69495916361248507062510162911, −3.39922203508551278265750534402, −2.10529219269452426976301251985, −0.53028133422494789623272081034,
0.71994524711808822694216853862, 1.63080067659525151140184494989, 2.84063955867562631955889535701, 3.33666485688659647234169149597, 4.78125141767371061765411102377, 5.42665153799671028022955382206, 6.60366449599131424889408544650, 6.85184382550724890585914285336, 7.55324969425668054068370311911, 8.242352116719530290875593952808