| L(s) = 1 | + 1.17·2-s − 1.17·3-s − 0.618·4-s + 1.23·5-s − 1.38·6-s − 4.23·7-s − 3.07·8-s − 1.61·9-s + 1.45·10-s − 3.61·11-s + 0.726·12-s + 3.07·13-s − 4.97·14-s − 1.45·15-s − 2.38·16-s + 2.47·17-s − 1.90·18-s − 0.763·20-s + 4.97·21-s − 4.25·22-s − 2.61·23-s + 3.61·24-s − 3.47·25-s + 3.61·26-s + 5.42·27-s + 2.61·28-s − 2.62·29-s + ⋯ |
| L(s) = 1 | + 0.831·2-s − 0.678·3-s − 0.309·4-s + 0.552·5-s − 0.564·6-s − 1.60·7-s − 1.08·8-s − 0.539·9-s + 0.459·10-s − 1.09·11-s + 0.209·12-s + 0.853·13-s − 1.33·14-s − 0.375·15-s − 0.595·16-s + 0.599·17-s − 0.448·18-s − 0.170·20-s + 1.08·21-s − 0.906·22-s − 0.545·23-s + 0.738·24-s − 0.694·25-s + 0.709·26-s + 1.04·27-s + 0.494·28-s − 0.488·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 3 | \( 1 + 1.17T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 6.43T + 37T^{2} \) |
| 41 | \( 1 - 9.23T + 41T^{2} \) |
| 43 | \( 1 + 6.32T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 0.449T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 + 0.527T + 61T^{2} \) |
| 67 | \( 1 + 0.171T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 - 1.23T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−11.06402681277488933976408638370, −10.05558877090894494927164344572, −9.356540448013613219754397813107, −8.224510779719056415531142987189, −6.62087999974218565435303278894, −5.81855294021072840271440899038, −5.37857801382705362187067334297, −3.80026632913059839285128960786, −2.83818145584408476715326291747, 0,
2.83818145584408476715326291747, 3.80026632913059839285128960786, 5.37857801382705362187067334297, 5.81855294021072840271440899038, 6.62087999974218565435303278894, 8.224510779719056415531142987189, 9.356540448013613219754397813107, 10.05558877090894494927164344572, 11.06402681277488933976408638370
