L(s) = 1 | + (0.933 − 1.45i)3-s + (−1.23 + 2.13i)5-s + (−0.5 − 0.866i)7-s + (−1.25 − 2.72i)9-s + (2.32 + 4.02i)11-s + (−3.55 + 6.15i)13-s + (1.96 + 3.78i)15-s − 4.51·17-s − 4.32·19-s + (−1.73 − 0.0789i)21-s + (2.93 − 5.08i)23-s + (−0.527 − 0.912i)25-s + (−5.14 − 0.708i)27-s + (3.48 + 6.04i)29-s + (−3.69 + 6.39i)31-s + ⋯ |
L(s) = 1 | + (0.538 − 0.842i)3-s + (−0.550 + 0.952i)5-s + (−0.188 − 0.327i)7-s + (−0.419 − 0.907i)9-s + (0.700 + 1.21i)11-s + (−0.985 + 1.70i)13-s + (0.506 + 0.977i)15-s − 1.09·17-s − 0.992·19-s + (−0.377 − 0.0172i)21-s + (0.611 − 1.05i)23-s + (−0.105 − 0.182i)25-s + (−0.990 − 0.136i)27-s + (0.647 + 1.12i)29-s + (−0.662 + 1.14i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0832 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.005167100\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005167100\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.933 + 1.45i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (1.23 - 2.13i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.32 - 4.02i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.55 - 6.15i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 + 4.32T + 19T^{2} \) |
| 23 | \( 1 + (-2.93 + 5.08i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.48 - 6.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.69 - 6.39i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 0.726T + 37T^{2} \) |
| 41 | \( 1 + (0.136 - 0.236i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.41 + 4.18i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.83 - 3.18i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 5.05T + 53T^{2} \) |
| 59 | \( 1 + (-4.56 + 7.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.90 - 11.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.663 - 1.14i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 4.32T + 73T^{2} \) |
| 79 | \( 1 + (-3.21 - 5.57i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.742 - 1.28i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9.83T + 89T^{2} \) |
| 97 | \( 1 + (-0.246 - 0.426i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.19628576573309304785478519347, −9.105374679360338258680061324168, −8.666568299671008844981208028562, −7.20491689322588801442120698906, −6.93670597072540966677283785116, −6.60255952326655432965003884455, −4.66893205225040431710220052693, −3.92161640037118546323911048747, −2.65689305551791856149339773300, −1.79749541168992469916997131788,
0.40774962780391401145471426510, 2.45797617606362914769964708672, 3.50130075459037375592109236441, 4.38947741893270277964301036358, 5.25086164311522622186102404553, 6.09548684183916287361136146872, 7.56674146338922983913008076537, 8.363105850909969329428678567778, 8.811922099553362245720731497925, 9.605734919808340039347473429599