L(s) = 1 | + (0.433 − 0.750i)5-s + (−2.25 + 1.38i)7-s + (1.75 + 3.04i)11-s − 1.86·13-s + (3.25 + 5.64i)17-s + (2.69 − 4.66i)19-s + (−4.32 + 7.48i)23-s + (2.12 + 3.67i)25-s + 3.51·29-s + (0.933 + 1.61i)31-s + (0.0576 + 2.29i)35-s + (1.39 − 2.40i)37-s + 10.3·41-s − 5.78·43-s + (−3.08 + 5.33i)47-s + ⋯ |
L(s) = 1 | + (0.193 − 0.335i)5-s + (−0.853 + 0.521i)7-s + (0.529 + 0.917i)11-s − 0.517·13-s + (0.790 + 1.36i)17-s + (0.617 − 1.06i)19-s + (−0.901 + 1.56i)23-s + (0.424 + 0.735i)25-s + 0.652·29-s + (0.167 + 0.290i)31-s + (0.00975 + 0.387i)35-s + (0.228 − 0.395i)37-s + 1.62·41-s − 0.881·43-s + (−0.449 + 0.778i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11156 + 0.709129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11156 + 0.709129i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.25 - 1.38i)T \) |
good | 5 | \( 1 + (-0.433 + 0.750i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.75 - 3.04i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 + (-3.25 - 5.64i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.32 - 7.48i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + (-0.933 - 1.61i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.39 + 2.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 + (3.08 - 5.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.80 + 4.85i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.82 - 4.89i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.14 - 8.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.676 - 1.17i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 + (3.62 + 6.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.83 - 10.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.86T + 83T^{2} \) |
| 89 | \( 1 + (-3.28 + 5.68i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.29T + 97T^{2} \) |
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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
−10.24393838693353919253744163808, −9.577256243752490064988343327843, −9.055839379598807959783633357841, −7.86815293402316976917740139305, −7.01669627709282047363259515343, −6.03733666333544540009299925074, −5.21656546488957749465779683935, −4.03751199991665491548538279779, −2.91635846889312121872544117964, −1.52818415336946036701435313843,
0.71169917092970086290171306817, 2.65599536862410957808295654032, 3.52263208261494885489982937587, 4.69190709367249105029218154987, 5.98129789091065574762051071701, 6.57047261099383844366658785280, 7.56228230358952010851077396590, 8.444353359579203390502137201244, 9.622930560300684250214245936148, 10.02575418518313827473226803569