L(s) = 1 | + (2.35 − 1.36i)2-s + (0.5 + 0.866i)3-s + (2.70 − 4.68i)4-s + 1.58i·5-s + (2.35 + 1.36i)6-s + (−0.866 − 0.5i)7-s − 9.26i·8-s + (−0.499 + 0.866i)9-s + (2.15 + 3.72i)10-s + (−4.79 + 2.77i)11-s + 5.40·12-s + (−1.34 + 3.34i)13-s − 2.72·14-s + (−1.37 + 0.791i)15-s + (−7.20 − 12.4i)16-s + (2.40 − 4.16i)17-s + ⋯ |
L(s) = 1 | + (1.66 − 0.962i)2-s + (0.288 + 0.499i)3-s + (1.35 − 2.34i)4-s + 0.707i·5-s + (0.962 + 0.555i)6-s + (−0.327 − 0.188i)7-s − 3.27i·8-s + (−0.166 + 0.288i)9-s + (0.680 + 1.17i)10-s + (−1.44 + 0.835i)11-s + 1.56·12-s + (−0.372 + 0.928i)13-s − 0.727·14-s + (−0.353 + 0.204i)15-s + (−1.80 − 3.11i)16-s + (0.582 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.71719 - 1.34257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.71719 - 1.34257i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (1.34 - 3.34i)T \) |
good | 2 | \( 1 + (-2.35 + 1.36i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 1.58iT - 5T^{2} \) |
| 11 | \( 1 + (4.79 - 2.77i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.40 + 4.16i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.772 - 0.446i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.31 + 4.00i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.941 - 1.63i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.47iT - 31T^{2} \) |
| 37 | \( 1 + (-4.32 + 2.49i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.45 - 4.87i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.506 + 0.877i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 12.5iT - 47T^{2} \) |
| 53 | \( 1 - 6.25T + 53T^{2} \) |
| 59 | \( 1 + (-2.98 - 1.72i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.79 + 3.10i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.7 + 7.37i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (11.3 + 6.56i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 5.33iT - 73T^{2} \) |
| 79 | \( 1 - 0.779T + 79T^{2} \) |
| 83 | \( 1 + 1.47iT - 83T^{2} \) |
| 89 | \( 1 + (-6.63 + 3.82i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.657 + 0.379i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.89261668566078836418775393773, −10.91599322798410492088012813778, −10.21390803795710241310597094834, −9.575978456717482703278306121680, −7.48207571821810702429236030062, −6.48672961415589771451950149702, −5.16726812602374416702273315418, −4.42398126035047326739070625252, −3.11055545909493115274141146038, −2.35216438680524340918771762880,
2.67669109315649238727974834335, 3.69861323246855202456510593678, 5.31533144455366152013608078348, 5.62833198717338760077453357934, 6.94301012493562220297007198887, 8.038688518402649053249211895784, 8.417531287306926402136886264660, 10.33742669728990178093766645580, 11.72540932755023456085980411506, 12.51350698249334912478175911190