| L(s) = 1 | − 2.33·2-s + 3.44·4-s + 1.04·5-s − 3.44·7-s − 3.38·8-s − 2.44·10-s − 5.71·11-s + 13-s + 8.05·14-s + 1.00·16-s − 2.09·17-s + 3.61·20-s + 13.3·22-s − 3.61·23-s − 3.89·25-s − 2.33·26-s − 11.8·28-s − 7.23·29-s + 9.44·31-s + 4.43·32-s + 4.89·34-s − 3.61·35-s − 3.89·37-s − 3.55·40-s − 9.33·41-s + 6.34·43-s − 19.7·44-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.72·4-s + 0.469·5-s − 1.30·7-s − 1.19·8-s − 0.774·10-s − 1.72·11-s + 0.277·13-s + 2.15·14-s + 0.250·16-s − 0.508·17-s + 0.809·20-s + 2.84·22-s − 0.754·23-s − 0.779·25-s − 0.457·26-s − 2.24·28-s − 1.34·29-s + 1.69·31-s + 0.783·32-s + 0.840·34-s − 0.611·35-s − 0.640·37-s − 0.561·40-s − 1.45·41-s + 0.968·43-s − 2.97·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3113639205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3113639205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 5 | \( 1 - 1.04T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 - 9.44T + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 + 9.33T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 + 9.33T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 - 7.81T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 0.348T + 67T^{2} \) |
| 71 | \( 1 - 7.23T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 0.348T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 5.24T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−8.620741741674488294125307360310, −8.101662386889603881218257165620, −7.36504445055386583799390930312, −6.57611415728692466507610322800, −5.98509219899833061110941301388, −5.02977840236687080566239802586, −3.63221061197768209613986895158, −2.61636869543808697411284817267, −1.94305790294613856791149466574, −0.40147547535220188110549982380,
0.40147547535220188110549982380, 1.94305790294613856791149466574, 2.61636869543808697411284817267, 3.63221061197768209613986895158, 5.02977840236687080566239802586, 5.98509219899833061110941301388, 6.57611415728692466507610322800, 7.36504445055386583799390930312, 8.101662386889603881218257165620, 8.620741741674488294125307360310
