L(s) = 1 | + (−2.14 + 0.575i)2-s − i·3-s + (2.54 − 1.46i)4-s + (3.44 + 0.922i)5-s + (0.575 + 2.14i)6-s + (2.25 − 1.38i)7-s + (−1.47 + 1.47i)8-s − 9-s − 7.92·10-s + (−2.44 + 2.44i)11-s + (−1.46 − 2.54i)12-s + (−0.668 − 3.54i)13-s + (−4.04 + 4.26i)14-s + (0.922 − 3.44i)15-s + (−0.624 + 1.08i)16-s + (1.26 + 2.18i)17-s + ⋯ |
L(s) = 1 | + (−1.51 + 0.406i)2-s − 0.577i·3-s + (1.27 − 0.734i)4-s + (1.53 + 0.412i)5-s + (0.234 + 0.876i)6-s + (0.852 − 0.522i)7-s + (−0.520 + 0.520i)8-s − 0.333·9-s − 2.50·10-s + (−0.736 + 0.736i)11-s + (−0.423 − 0.734i)12-s + (−0.185 − 0.982i)13-s + (−1.08 + 1.13i)14-s + (0.238 − 0.889i)15-s + (−0.156 + 0.270i)16-s + (0.306 + 0.530i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.835429 - 0.0611772i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.835429 - 0.0611772i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (-2.25 + 1.38i)T \) |
| 13 | \( 1 + (0.668 + 3.54i)T \) |
good | 2 | \( 1 + (2.14 - 0.575i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-3.44 - 0.922i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (2.44 - 2.44i)T - 11iT^{2} \) |
| 17 | \( 1 + (-1.26 - 2.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.15 + 3.15i)T - 19iT^{2} \) |
| 23 | \( 1 + (-2.64 - 1.52i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.12 + 1.95i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.54 + 5.75i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-2.71 - 10.1i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.06 + 0.553i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.23 - 1.86i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.75 + 6.55i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.54 - 4.40i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.20 + 8.21i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 13.0iT - 61T^{2} \) |
| 67 | \( 1 + (6.52 + 6.52i)T + 67iT^{2} \) |
| 71 | \( 1 + (12.8 - 3.43i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.73 + 1.26i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (4.45 + 7.72i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.573 + 0.573i)T - 83iT^{2} \) |
| 89 | \( 1 + (8.03 - 2.15i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-4.06 - 15.1i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.47812492696068933090208086547, −10.41817707141053494549877465530, −10.05039331158355723141693485464, −9.033614247425143758438508540980, −7.84325874203870289720875663648, −7.34798245320435004115150785998, −6.22275979514060105134201898779, −5.16021338010942728979842447767, −2.47877591054106652199791497896, −1.29689931535208063265309794970,
1.51850790119485471859492205164, 2.66840033409664014527611550316, 4.98163262251146857239219783334, 5.79149646604454200905654612668, 7.40813215469738405811921752653, 8.621535134547815451376010559191, 9.125450843750067114629211130785, 9.864803729849663947678438329721, 10.71464065963005260364924615959, 11.44528820100635438984428923703