L(s) = 1 | + 2·4-s + (−2 − 1.73i)7-s + (−1 − 3.46i)13-s + 4·16-s + (7.5 − 4.33i)19-s + (−2.5 − 4.33i)25-s + (−4 − 3.46i)28-s + (7.5 − 4.33i)31-s + 5.19i·37-s + (−4 + 6.92i)43-s + (1.00 + 6.92i)49-s + (−2 − 6.92i)52-s + (−0.5 − 0.866i)61-s + 8·64-s + (−10.5 − 6.06i)67-s + ⋯ |
L(s) = 1 | + 4-s + (−0.755 − 0.654i)7-s + (−0.277 − 0.960i)13-s + 16-s + (1.72 − 0.993i)19-s + (−0.5 − 0.866i)25-s + (−0.755 − 0.654i)28-s + (1.34 − 0.777i)31-s + 0.854i·37-s + (−0.609 + 1.05i)43-s + (0.142 + 0.989i)49-s + (−0.277 − 0.960i)52-s + (−0.0640 − 0.110i)61-s + 64-s + (−1.28 − 0.740i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60651 - 0.781725i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60651 - 0.781725i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 2 | \( 1 - 2T^{2} \) |
| 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (-7.5 + 4.33i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.5 + 4.33i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.19iT - 37T^{2} \) |
| 41 | \( 1 + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.5 + 6.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12 + 6.92i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.03090772372242502431819257984, −9.635023765519565968415240702370, −8.142200196690630770983997233034, −7.50071049439439988675137716864, −6.67140302187334630284381890088, −5.91367053766579945903153450923, −4.76036566512188688452287553418, −3.34539734953464998389171891832, −2.66819301919653587642606797314, −0.917135504782075917951374909555,
1.61389568997971843237903016251, 2.80543961373722632397572010385, 3.69957392534034410954236123761, 5.27324062478636023442147678404, 6.03362407839614074902443083005, 6.92177530320279608055596495996, 7.59583823397693846133706873422, 8.723162376681562215409651022055, 9.661705528373869945269933876820, 10.21542044498469552153961220043