L(s) = 1 | + (−0.233 + 1.71i)3-s + (−0.0868 − 0.150i)5-s + (−2.60 − 0.486i)7-s + (−2.89 − 0.799i)9-s + (3.25 + 1.87i)11-s + (−3.54 + 2.04i)13-s + (0.278 − 0.114i)15-s − 6.00·17-s + 6.26i·19-s + (1.44 − 4.34i)21-s + (−6.37 + 3.68i)23-s + (2.48 − 4.30i)25-s + (2.04 − 4.77i)27-s + (−8.58 − 4.95i)29-s + (3.76 − 2.17i)31-s + ⋯ |
L(s) = 1 | + (−0.134 + 0.990i)3-s + (−0.0388 − 0.0672i)5-s + (−0.982 − 0.184i)7-s + (−0.963 − 0.266i)9-s + (0.980 + 0.566i)11-s + (−0.982 + 0.567i)13-s + (0.0718 − 0.0294i)15-s − 1.45·17-s + 1.43i·19-s + (0.314 − 0.949i)21-s + (−1.32 + 0.767i)23-s + (0.496 − 0.860i)25-s + (0.393 − 0.919i)27-s + (−1.59 − 0.920i)29-s + (0.676 − 0.390i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0285168 + 0.537811i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0285168 + 0.537811i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.233 - 1.71i)T \) |
| 7 | \( 1 + (2.60 + 0.486i)T \) |
good | 5 | \( 1 + (0.0868 + 0.150i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.25 - 1.87i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.54 - 2.04i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6.00T + 17T^{2} \) |
| 19 | \( 1 - 6.26iT - 19T^{2} \) |
| 23 | \( 1 + (6.37 - 3.68i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (8.58 + 4.95i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.76 + 2.17i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.23T + 37T^{2} \) |
| 41 | \( 1 + (0.489 + 0.848i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.0468 + 0.0811i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.86 - 3.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 5.61iT - 53T^{2} \) |
| 59 | \( 1 + (-0.620 - 1.07i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.45 + 4.30i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.21 - 7.30i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 - 15.4iT - 73T^{2} \) |
| 79 | \( 1 + (1.21 - 2.10i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.16 + 5.47i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + (-5.02 - 2.90i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.41022381802690414563803266259, −10.13958124699828298308043488185, −9.698886078633618710249017215984, −9.040078429517097475821393174330, −7.77981295548243611472807150482, −6.56755446889878189888152999906, −5.85985093290122254076031310619, −4.33247266228511180079703382577, −3.93924198868731460740493474192, −2.34060772880096061178586168453,
0.30288662029801301917592388230, 2.22849330691342522244675733254, 3.31973964387625514361828000945, 4.88004771400862255254137427618, 6.17795636619968966825043889955, 6.70387087757283452442620595600, 7.58850210860621649605485821270, 8.828168226509457610399914378381, 9.338642609965590045997146805440, 10.68555996436532027247597634853