L(s) = 1 | + (−3.27 − 1.89i)11-s − 8.02i·17-s + 8.34·19-s + (2.5 − 4.33i)25-s + (0.398 − 0.230i)41-s + (−1.17 + 2.03i)43-s + (−3.5 − 6.06i)49-s + (10.6 − 6.13i)59-s + (−7.17 − 12.4i)67-s + 13.6·73-s + (−2.44 − 1.41i)83-s + 5.65i·89-s + (−9.84 + 17.0i)97-s − 4.70i·107-s + (−9.79 + 5.65i)113-s + ⋯ |
L(s) = 1 | + (−0.987 − 0.570i)11-s − 1.94i·17-s + 1.91·19-s + (0.5 − 0.866i)25-s + (0.0623 − 0.0359i)41-s + (−0.179 + 0.310i)43-s + (−0.5 − 0.866i)49-s + (1.38 − 0.798i)59-s + (−0.876 − 1.51i)67-s + 1.60·73-s + (−0.268 − 0.155i)83-s + 0.599i·89-s + (−0.999 + 1.73i)97-s − 0.454i·107-s + (−0.921 + 0.532i)113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15004 - 0.728934i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15004 - 0.728934i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.27 + 1.89i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 8.02iT - 17T^{2} \) |
| 19 | \( 1 - 8.34T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (-0.398 + 0.230i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.17 - 2.03i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-10.6 + 6.13i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.17 + 12.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.44 + 1.41i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.65iT - 89T^{2} \) |
| 97 | \( 1 + (9.84 - 17.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.894698935907858296858069353309, −9.322061658197877863940058419151, −8.226139525429808086681492738152, −7.51289408425471608124750486816, −6.65578894929035871887264689745, −5.38808268541079383633326944895, −4.92248920510455505575774375336, −3.37572739226034554294294729497, −2.56558931867971276934818502418, −0.70359380349248619104052092130,
1.47204781867105401788957432602, 2.85814276036973291429470768753, 3.93594543018270334249324761742, 5.12405992319947951324973678043, 5.82253065247193381760125788530, 7.02633632569804065955786419058, 7.75356880050843623520001103169, 8.564993072904004637242555315115, 9.596197586606250809916012474307, 10.28466486575814716285276580168