| L(s) = 1 | + (−0.654 − 0.755i)2-s + (−1.39 − 0.899i)3-s + (−0.142 + 0.989i)4-s + (−0.109 + 0.240i)5-s + (0.236 + 1.64i)6-s + (−0.959 − 0.281i)7-s + (0.841 − 0.540i)8-s + (−0.0958 − 0.209i)9-s + (0.253 − 0.0744i)10-s + (−2.07 + 2.39i)11-s + (1.08 − 1.25i)12-s + (−2.60 + 0.763i)13-s + (0.415 + 0.909i)14-s + (0.369 − 0.237i)15-s + (−0.959 − 0.281i)16-s + (0.917 + 6.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.808 − 0.519i)3-s + (−0.0711 + 0.494i)4-s + (−0.0490 + 0.107i)5-s + (0.0966 + 0.672i)6-s + (−0.362 − 0.106i)7-s + (0.297 − 0.191i)8-s + (−0.0319 − 0.0699i)9-s + (0.0801 − 0.0235i)10-s + (−0.624 + 0.720i)11-s + (0.314 − 0.363i)12-s + (−0.721 + 0.211i)13-s + (0.111 + 0.243i)14-s + (0.0955 − 0.0613i)15-s + (−0.239 − 0.0704i)16-s + (0.222 + 1.54i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0371 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0371 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.181897 + 0.175261i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.181897 + 0.175261i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (1.77 + 4.45i)T \) |
| good | 3 | \( 1 + (1.39 + 0.899i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (0.109 - 0.240i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (2.07 - 2.39i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.60 - 0.763i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.917 - 6.38i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.374 - 2.60i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.829 - 5.77i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.67 + 1.07i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.01 + 6.60i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (2.94 - 6.43i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (4.71 + 3.02i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 + (-4.06 - 1.19i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (14.1 - 4.14i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.01 + 1.29i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (8.51 + 9.83i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (2.00 + 2.31i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.28 + 8.94i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-4.32 + 1.26i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.99 - 8.75i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (3.26 + 2.09i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.75 + 6.02i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−12.01566343113542042035104211451, −10.77242525208435719522882449791, −10.28695886982137932086984835511, −9.180318554637685082926775384809, −8.031818452959025672473953343864, −7.03890280800589998803886089140, −6.15385153675579789478581041537, −4.84187910480056291007752234729, −3.35909166587354703206768323701, −1.73285795094124373316138406834,
0.22429037778590733452971557784, 2.83502333826485146626322714302, 4.76174693156926731436826397786, 5.38560704476274475323757057406, 6.45958114596642074927390303611, 7.57530462003096477536854114807, 8.540963287713247786597562489020, 9.725097347777080927850025193034, 10.26936641548298217253180529638, 11.34513958873916864822317529008
