L(s) = 1 | − 3.72i·2-s − 5.84·4-s + (−1.33 − 2.30i)5-s + (8.98 + 16.1i)7-s − 8.01i·8-s + (−8.57 + 4.94i)10-s + (−58.8 − 33.9i)11-s + (−69.6 − 40.2i)13-s + (60.2 − 33.4i)14-s − 76.5·16-s + (29.6 + 51.2i)17-s + (−5.86 − 3.38i)19-s + (7.77 + 13.4i)20-s + (−126. + 219. i)22-s + (−109. + 63.2i)23-s + ⋯ |
L(s) = 1 | − 1.31i·2-s − 0.730·4-s + (−0.118 − 0.206i)5-s + (0.485 + 0.874i)7-s − 0.354i·8-s + (−0.271 + 0.156i)10-s + (−1.61 − 0.931i)11-s + (−1.48 − 0.858i)13-s + (1.15 − 0.638i)14-s − 1.19·16-s + (0.422 + 0.731i)17-s + (−0.0707 − 0.0408i)19-s + (0.0869 + 0.150i)20-s + (−1.22 + 2.12i)22-s + (−0.992 + 0.572i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.288871 + 0.667402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.288871 + 0.667402i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-8.98 - 16.1i)T \) |
good | 2 | \( 1 + 3.72iT - 8T^{2} \) |
| 5 | \( 1 + (1.33 + 2.30i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (58.8 + 33.9i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (69.6 + 40.2i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-29.6 - 51.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (5.86 + 3.38i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (109. - 63.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-114. + 66.0i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 72.5iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (63.8 - 110. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-4.93 + 8.54i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (108. + 188. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 185.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (174. - 100. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 178.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 225. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 438.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 533. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-287. + 165. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (590. + 1.02e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-66.3 + 114. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-761. + 439. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.58918950280620264499788032664, −10.48115944232414676855736181406, −9.969995576231228407613470734627, −8.528120931668056561927307227653, −7.73532595163567828864256112210, −5.83027298951124244959142164501, −4.78454399383585802695636860415, −3.09008689847347814934849598347, −2.20513767161883605247541153677, −0.29398703964626688357686581060,
2.39166611169688723026605548445, 4.59717228540215416157258042686, 5.24034321558652689110722690924, 6.94487646553114203895491299979, 7.36859621161865560913388604707, 8.213901711978636965568974737437, 9.703496447053793983995153552591, 10.59587481867916970129825135455, 11.79786469210714069873464962833, 12.91771798954306074585332406112