| L(s) = 1 | + (−0.776 − 1.18i)2-s + (−0.846 + 1.51i)3-s + (−0.794 + 1.83i)4-s + (0.448 + 2.19i)5-s + (2.44 − 0.172i)6-s − 2.41i·7-s + (2.78 − 0.486i)8-s + (−1.56 − 2.55i)9-s + (2.24 − 2.23i)10-s + (−4.39 + 3.19i)11-s + (−2.10 − 2.75i)12-s + (3.38 + 2.45i)13-s + (−2.85 + 1.87i)14-s + (−3.68 − 1.17i)15-s + (−2.73 − 2.91i)16-s + (−3.42 + 1.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.549 − 0.835i)2-s + (−0.488 + 0.872i)3-s + (−0.397 + 0.917i)4-s + (0.200 + 0.979i)5-s + (0.997 − 0.0703i)6-s − 0.913i·7-s + (0.985 − 0.171i)8-s + (−0.522 − 0.852i)9-s + (0.708 − 0.705i)10-s + (−1.32 + 0.963i)11-s + (−0.606 − 0.795i)12-s + (0.937 + 0.681i)13-s + (−0.763 + 0.501i)14-s + (−0.952 − 0.303i)15-s + (−0.684 − 0.729i)16-s + (−0.830 + 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.220456 + 0.394989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.220456 + 0.394989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.776 + 1.18i)T \) |
| 3 | \( 1 + (0.846 - 1.51i)T \) |
| 5 | \( 1 + (-0.448 - 2.19i)T \) |
| good | 7 | \( 1 + 2.41iT - 7T^{2} \) |
| 11 | \( 1 + (4.39 - 3.19i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.38 - 2.45i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.42 - 1.11i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (7.02 - 2.28i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.603 - 0.438i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.66 + 1.19i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.927 + 0.301i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.42 - 3.21i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.05 + 1.45i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 4.88iT - 43T^{2} \) |
| 47 | \( 1 + (-2.13 + 6.58i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (4.55 + 1.48i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.42 - 5.39i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.05 - 5.85i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.52 + 0.495i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.21 - 9.88i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.45 + 2.51i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-8.43 - 2.74i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.68 + 8.27i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.10 - 9.77i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.0838 + 0.258i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.59362012368734282506490020320, −10.76617987689495107975731396639, −10.46398571480612665210930473698, −9.678950685272267853439060225645, −8.483750279094843532752478336190, −7.30939866041320323990109898211, −6.23668962473461562976445712429, −4.50998397955268032159633514860, −3.76770422178236877987742423218, −2.25330614231277288162337379518,
0.40225569089088572876217349162, 2.20453501028644597330104356425, 4.84379769068483842756933290454, 5.74686020967039120785750579029, 6.26868869784182392855392191805, 7.78799515596055598810779621746, 8.470251697910478324045925622438, 9.035025228740017840974761197973, 10.62727411111040872966641091446, 11.21231570346832119655996267616
