| L(s) = 1 | + 2.70·2-s + 5.31·4-s − 3.15·5-s + 1.33·7-s + 8.95·8-s − 8.51·10-s + 4.59·11-s + 2.82·13-s + 3.59·14-s + 13.5·16-s − 4.70·17-s − 8.17·19-s − 16.7·20-s + 12.4·22-s − 4.03·23-s + 4.92·25-s + 7.63·26-s + 7.06·28-s + 2.28·29-s + 3.87·31-s + 18.8·32-s − 12.7·34-s − 4.19·35-s − 3.85·37-s − 22.1·38-s − 28.2·40-s + 0.380·41-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.65·4-s − 1.40·5-s + 0.502·7-s + 3.16·8-s − 2.69·10-s + 1.38·11-s + 0.782·13-s + 0.961·14-s + 3.39·16-s − 1.13·17-s − 1.87·19-s − 3.74·20-s + 2.64·22-s − 0.840·23-s + 0.985·25-s + 1.49·26-s + 1.33·28-s + 0.423·29-s + 0.696·31-s + 3.33·32-s − 2.17·34-s − 0.708·35-s − 0.633·37-s − 3.58·38-s − 4.46·40-s + 0.0594·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.971107277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.971107277\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.70T + 17T^{2} \) |
| 19 | \( 1 + 8.17T + 19T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 - 2.28T + 29T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 + 3.85T + 37T^{2} \) |
| 41 | \( 1 - 0.380T + 41T^{2} \) |
| 43 | \( 1 + 3.97T + 43T^{2} \) |
| 47 | \( 1 + 5.73T + 47T^{2} \) |
| 53 | \( 1 + 1.81T + 53T^{2} \) |
| 59 | \( 1 - 1.82T + 59T^{2} \) |
| 67 | \( 1 + 9.39T + 67T^{2} \) |
| 71 | \( 1 + 5.89T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 4.87T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−11.27594013984104239288125616430, −10.57296827559257434981889404365, −8.681840091325019894549877766463, −7.902883182737579700829796301807, −6.64014548732183586754832243684, −6.33287992778663176742438461267, −4.70287241209355048159564174179, −4.18521803090260859881304562880, −3.48903214165376976403830790023, −1.90968587964786898377356030839,
1.90968587964786898377356030839, 3.48903214165376976403830790023, 4.18521803090260859881304562880, 4.70287241209355048159564174179, 6.33287992778663176742438461267, 6.64014548732183586754832243684, 7.902883182737579700829796301807, 8.681840091325019894549877766463, 10.57296827559257434981889404365, 11.27594013984104239288125616430
