L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (1.94 − 3.37i)5-s + 7.99·8-s + (3.89 + 6.75i)10-s + (−30.6 − 53.1i)11-s − 53.6·13-s + (−8 + 13.8i)16-s + (−16.0 − 27.7i)17-s + (−27.8 + 48.3i)19-s − 15.5·20-s + 122.·22-s + (−47.3 + 81.9i)23-s + (54.8 + 95.0i)25-s + (53.6 − 93.0i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.174 − 0.302i)5-s + 0.353·8-s + (0.123 + 0.213i)10-s + (−0.841 − 1.45i)11-s − 1.14·13-s + (−0.125 + 0.216i)16-s + (−0.228 − 0.396i)17-s + (−0.336 + 0.583i)19-s − 0.174·20-s + 1.18·22-s + (−0.428 + 0.742i)23-s + (0.439 + 0.760i)25-s + (0.405 − 0.701i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9477732162\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9477732162\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.94 + 3.37i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (30.6 + 53.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 53.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (16.0 + 27.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (27.8 - 48.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (47.3 - 81.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-66.3 - 114. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (74.6 - 129. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 427.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (28.5 - 49.3i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (131. + 228. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (225. + 391. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-289. + 501. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (154. + 268. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-596. - 1.03e3i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (659. - 1.14e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-116. + 201. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−9.708105594945329745017685644741, −9.123598931589330606431319864171, −8.120389950164147583090808016070, −7.60364134333680865073358265895, −6.50994135632485623020398425925, −5.55600031177280125140090601833, −4.99971221706584919363118943696, −3.60468471299437802519719356598, −2.34432736295477954462089887187, −0.819270616589481615502111481695,
0.36357337352415531894280859600, 2.18269859558839755151234832493, 2.55018968910184049537498189922, 4.19243534058106496415949805305, 4.83971511523965663374739042027, 6.12489679631917629966206551459, 7.29883981885522250814564945099, 7.72734829137358865652338914359, 8.952494554967878068254215958780, 9.671186364710946874117563628692