| L(s) = 1 | − 2.52·2-s + 4.37·4-s − 0.792·5-s + 7-s − 5.98·8-s + 2·10-s + 3.46·11-s − 13-s − 2.52·14-s + 6.37·16-s − 5.04·17-s + 7.37·19-s − 3.46·20-s − 8.74·22-s + 7.72·23-s − 4.37·25-s + 2.52·26-s + 4.37·28-s − 2.67·29-s + 0.627·31-s − 4.10·32-s + 12.7·34-s − 0.792·35-s − 4.74·37-s − 18.6·38-s + 4.74·40-s + 5.04·41-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 2.18·4-s − 0.354·5-s + 0.377·7-s − 2.11·8-s + 0.632·10-s + 1.04·11-s − 0.277·13-s − 0.674·14-s + 1.59·16-s − 1.22·17-s + 1.69·19-s − 0.774·20-s − 1.86·22-s + 1.60·23-s − 0.874·25-s + 0.495·26-s + 0.826·28-s − 0.496·29-s + 0.112·31-s − 0.726·32-s + 2.18·34-s − 0.133·35-s − 0.780·37-s − 3.01·38-s + 0.750·40-s + 0.788·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6741204829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6741204829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 + 0.792T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 - 7.72T + 23T^{2} \) |
| 29 | \( 1 + 2.67T + 29T^{2} \) |
| 31 | \( 1 - 0.627T + 31T^{2} \) |
| 37 | \( 1 + 4.74T + 37T^{2} \) |
| 41 | \( 1 - 5.04T + 41T^{2} \) |
| 43 | \( 1 + 6.11T + 43T^{2} \) |
| 47 | \( 1 - 7.42T + 47T^{2} \) |
| 53 | \( 1 + 4.25T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 2.74T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 + 0.792T + 89T^{2} \) |
| 97 | \( 1 - 9.37T + 97T^{2} \) |
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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
−10.01385536143211449478842930487, −9.178570825341107463438376197276, −8.832578002310324463196402022984, −7.70641928509700399162404682442, −7.19226838700169198161264631546, −6.33866200099882094852895118711, −4.95925489887696130245337990565, −3.49895641740204013731155918909, −2.10692426235427251914629193979, −0.894204021139868532493956564418,
0.894204021139868532493956564418, 2.10692426235427251914629193979, 3.49895641740204013731155918909, 4.95925489887696130245337990565, 6.33866200099882094852895118711, 7.19226838700169198161264631546, 7.70641928509700399162404682442, 8.832578002310324463196402022984, 9.178570825341107463438376197276, 10.01385536143211449478842930487
