| L(s) = 1 | − 1.32·2-s − 1.38·3-s − 0.232·4-s − 0.595·5-s + 1.83·6-s − 3.72·7-s + 2.96·8-s − 1.09·9-s + 0.791·10-s − 4.08·11-s + 0.320·12-s − 0.436·13-s + 4.95·14-s + 0.822·15-s − 3.48·16-s + 17-s + 1.45·18-s − 4.36·19-s + 0.138·20-s + 5.14·21-s + 5.42·22-s − 3.77·23-s − 4.09·24-s − 4.64·25-s + 0.579·26-s + 5.65·27-s + 0.864·28-s + ⋯ |
| L(s) = 1 | − 0.940·2-s − 0.797·3-s − 0.116·4-s − 0.266·5-s + 0.749·6-s − 1.40·7-s + 1.04·8-s − 0.364·9-s + 0.250·10-s − 1.23·11-s + 0.0925·12-s − 0.120·13-s + 1.32·14-s + 0.212·15-s − 0.870·16-s + 0.242·17-s + 0.342·18-s − 1.00·19-s + 0.0309·20-s + 1.12·21-s + 1.15·22-s − 0.787·23-s − 0.836·24-s − 0.929·25-s + 0.113·26-s + 1.08·27-s + 0.163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1809271170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1809271170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.32T + 2T^{2} \) |
| 3 | \( 1 + 1.38T + 3T^{2} \) |
| 5 | \( 1 + 0.595T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 + 4.08T + 11T^{2} \) |
| 13 | \( 1 + 0.436T + 13T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 23 | \( 1 + 3.77T + 23T^{2} \) |
| 29 | \( 1 - 0.0165T + 29T^{2} \) |
| 31 | \( 1 - 8.31T + 31T^{2} \) |
| 37 | \( 1 + 3.71T + 37T^{2} \) |
| 41 | \( 1 - 3.97T + 41T^{2} \) |
| 47 | \( 1 + 6.88T + 47T^{2} \) |
| 53 | \( 1 - 6.96T + 53T^{2} \) |
| 59 | \( 1 - 7.32T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 0.915T + 67T^{2} \) |
| 71 | \( 1 - 0.304T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−10.09128824240246719516834770150, −9.874562014959677948642255124506, −8.595334222876303671895168084776, −8.035898704171562161379703589650, −6.92547265335538385833499654262, −6.05660980678262374861144951223, −5.11807697596727217388274548548, −3.91734883433976788346825329878, −2.52527188415534827572984840577, −0.39902228955380175252813270664,
0.39902228955380175252813270664, 2.52527188415534827572984840577, 3.91734883433976788346825329878, 5.11807697596727217388274548548, 6.05660980678262374861144951223, 6.92547265335538385833499654262, 8.035898704171562161379703589650, 8.595334222876303671895168084776, 9.874562014959677948642255124506, 10.09128824240246719516834770150
