L(s) = 1 | + (−0.5 + 0.866i)7-s + (1.5 − 2.59i)11-s + (−0.5 − 0.866i)13-s + 6·17-s − 4·19-s + (1.5 + 2.59i)23-s + (1.5 − 2.59i)29-s + (−2.5 − 4.33i)31-s − 2·37-s + (1.5 + 2.59i)41-s + (−0.5 + 0.866i)43-s + (4.5 − 7.79i)47-s + (3 + 5.19i)49-s − 6·53-s + (−1.5 − 2.59i)59-s + ⋯ |
L(s) = 1 | + (−0.188 + 0.327i)7-s + (0.452 − 0.783i)11-s + (−0.138 − 0.240i)13-s + 1.45·17-s − 0.917·19-s + (0.312 + 0.541i)23-s + (0.278 − 0.482i)29-s + (−0.449 − 0.777i)31-s − 0.328·37-s + (0.234 + 0.405i)41-s + (−0.0762 + 0.132i)43-s + (0.656 − 1.13i)47-s + (0.428 + 0.742i)49-s − 0.824·53-s + (−0.195 − 0.338i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.727640417\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.727640417\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-5.5 + 9.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.700059130137567478704344729243, −8.059132000704741858442902833906, −7.30703586745034570431967184093, −6.29883229225748573767382195166, −5.77748498234514986135667973289, −4.92450715109776756216432577419, −3.79850725971616067001394431616, −3.14594194224366965848176516650, −1.99513772462782543979242594603, −0.67656355528586409631354146224,
1.05141052407344781931165861592, 2.19135555622565995720201579620, 3.32143267160992003618237791114, 4.15319126485916762715176826787, 4.96931322002226625047524600859, 5.86223857572940461729381863652, 6.79755045385266710969987275832, 7.26759846298857389407542883761, 8.177541086239667217473287099925, 8.955093213528559743554773459310