| L(s) = 1 | + (1.86 − 0.328i)2-s + (−1.24 − 1.48i)3-s + (−0.390 + 0.142i)4-s + (2.62 + 0.955i)5-s + (−2.81 − 2.36i)6-s + (−1.20 + 2.09i)7-s + (−7.23 + 4.17i)8-s + (0.907 − 5.14i)9-s + (5.20 + 0.918i)10-s + (−9.60 − 16.6i)11-s + (0.699 + 0.403i)12-s + (9.33 − 11.1i)13-s + (−1.56 + 4.29i)14-s + (−1.85 − 5.09i)15-s + (−10.8 + 9.10i)16-s + (−2.97 − 16.8i)17-s + ⋯ |
| L(s) = 1 | + (0.932 − 0.164i)2-s + (−0.416 − 0.496i)3-s + (−0.0976 + 0.0355i)4-s + (0.524 + 0.191i)5-s + (−0.469 − 0.394i)6-s + (−0.172 + 0.298i)7-s + (−0.904 + 0.522i)8-s + (0.100 − 0.571i)9-s + (0.520 + 0.0918i)10-s + (−0.873 − 1.51i)11-s + (0.0582 + 0.0336i)12-s + (0.717 − 0.855i)13-s + (−0.111 + 0.307i)14-s + (−0.123 − 0.339i)15-s + (−0.678 + 0.569i)16-s + (−0.175 − 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.708910 - 1.36936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.708910 - 1.36936i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-1.86 + 0.328i)T + (3.75 - 1.36i)T^{2} \) |
| 3 | \( 1 + (1.24 + 1.48i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (-2.62 - 0.955i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (1.20 - 2.09i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (9.60 + 16.6i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-9.33 + 11.1i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (2.97 + 16.8i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (-7.16 + 2.60i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-8.11 - 1.43i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (5.19 + 3.00i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 59.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (19.8 + 23.6i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (23.0 + 8.40i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-12.7 + 72.0i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (-9.58 - 26.3i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (-9.07 + 1.60i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-57.9 + 21.0i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (11.8 + 2.09i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (13.1 - 36.1i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-38.9 + 32.6i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-69.0 - 82.3i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (35.6 - 61.7i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-5.21 + 6.21i)T + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-117. + 20.6i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.23684359447945606742908495825, −10.15041921775389690660196146533, −8.949329963763324565449024374867, −8.164156547215480213151281165517, −6.63515320888219357212646126537, −5.81456627216100955886898411229, −5.23406726338174817433690933914, −3.57941623643723884273053832814, −2.72466368905635049521649927451, −0.53318151757829559470278448289,
2.00741301076307608218763809343, 3.85632293164750263490566479507, 4.66704622838561841017531102218, 5.44552243520122288751343168006, 6.40937183257093931762987327065, 7.57325685760319476516343325276, 8.980310215695901322788679617582, 9.872864956135206140439039984137, 10.52671273042693615139680539169, 11.62674689511223004231736000188
