L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (0.190 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 0.618·10-s + (2.54 − 2.12i)11-s + 0.999·12-s + (−2 − 6.15i)13-s + (0.809 + 0.587i)14-s + (−0.5 + 0.363i)15-s + (0.309 − 0.951i)16-s + (−0.354 + 1.08i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.0854 − 0.262i)5-s + (0.126 − 0.388i)6-s + (0.305 − 0.222i)7-s + (−0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + 0.195·10-s + (0.767 − 0.641i)11-s + 0.288·12-s + (−0.554 − 1.70i)13-s + (0.216 + 0.157i)14-s + (−0.129 + 0.0937i)15-s + (0.0772 − 0.237i)16-s + (−0.0858 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32042 - 0.194281i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32042 - 0.194281i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-2.54 + 2.12i)T \) |
good | 5 | \( 1 + (-0.190 + 0.587i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (2 + 6.15i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.354 - 1.08i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.92 - 2.12i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 3.61T + 23T^{2} \) |
| 29 | \( 1 + (-3.85 + 2.80i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.80 + 8.64i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.35 + 4.61i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-8.35 - 6.06i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + (4.61 + 3.35i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-3 - 9.23i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (10.0 - 7.33i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.85 + 11.8i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + (3.70 - 11.4i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (12.0 - 8.78i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.236 + 0.726i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.76 + 14.6i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 5.14T + 89T^{2} \) |
| 97 | \( 1 + (-1.52 - 4.70i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.12318959238344882692284461809, −10.10473063762621457715750300032, −9.070542486563081961958107471584, −7.992084498417564312253891818724, −7.41914536495502405799971457176, −6.16623760046549232525674422471, −5.52167558654385592881398692068, −4.47296249641728526598337275123, −3.07747484074146045460086522563, −0.932030804009249173055129699981,
1.57811683828562166930062363741, 3.02592350621511411564294489224, 4.47591093323281900067620335310, 4.96066692486672050359356076451, 6.48073771798556650652968406000, 7.13579588075527485010325511451, 8.853124771406515062514893301894, 9.380824379285199886313836326281, 10.31173827253111516649180743180, 11.24469911134173111990769373150