| L(s) = 1 | + (0.260 + 0.300i)2-s + (−0.415 + 0.909i)3-s + (0.262 − 1.82i)4-s + (3.42 + 1.00i)5-s + (−0.381 + 0.112i)6-s + (−1.73 − 2.00i)7-s + (1.28 − 0.825i)8-s + (−0.654 − 0.755i)9-s + (0.589 + 1.29i)10-s + (−0.341 − 0.100i)11-s + (1.54 + 0.995i)12-s + (4.24 + 2.72i)13-s + (0.150 − 1.04i)14-s + (−2.33 + 2.69i)15-s + (−2.95 − 0.866i)16-s + (0.330 + 2.30i)17-s + ⋯ |
| L(s) = 1 | + (0.184 + 0.212i)2-s + (−0.239 + 0.525i)3-s + (0.131 − 0.911i)4-s + (1.53 + 0.450i)5-s + (−0.155 + 0.0457i)6-s + (−0.656 − 0.758i)7-s + (0.454 − 0.291i)8-s + (−0.218 − 0.251i)9-s + (0.186 + 0.408i)10-s + (−0.103 − 0.0302i)11-s + (0.447 + 0.287i)12-s + (1.17 + 0.755i)13-s + (0.0401 − 0.279i)14-s + (−0.603 + 0.696i)15-s + (−0.738 − 0.216i)16-s + (0.0802 + 0.558i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47147 + 0.109193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47147 + 0.109193i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (8.00 + 1.72i)T \) |
| good | 2 | \( 1 + (-0.260 - 0.300i)T + (-0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (-3.42 - 1.00i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (1.73 + 2.00i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (0.341 + 0.100i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-4.24 - 2.72i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.330 - 2.30i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (3.34 - 3.85i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.69 + 5.90i)T + (-15.0 - 17.3i)T^{2} \) |
| 29 | \( 1 + 4.53T + 29T^{2} \) |
| 31 | \( 1 + (3.84 - 2.47i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + (-0.433 - 3.01i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.946 + 6.58i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (4.91 - 10.7i)T + (-30.7 - 35.5i)T^{2} \) |
| 53 | \( 1 + (-0.998 + 6.94i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-8.18 + 5.25i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.52 - 0.448i)T + (51.3 - 32.9i)T^{2} \) |
| 71 | \( 1 + (-0.0536 + 0.373i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (0.0734 - 0.0215i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-13.1 - 8.43i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.16 - 0.929i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-6.03 - 13.2i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−12.78556565504754510572107657691, −10.93078636194509597746107108138, −10.51611073648552454020275001242, −9.800804644404483295389446312604, −8.839504668126984997597090844992, −6.69732906801020985277219772166, −6.31104952929352827630464293414, −5.28688262030549907824477781131, −3.78578663036265597212100103747, −1.77831758836140411407144763689,
1.97261387871071962963562557139, 3.22808016336395342993702949106, 5.23848610576789669407645888266, 6.06334183316310311816503121723, 7.19945905291341277139379592357, 8.649060905116552144925130129037, 9.254128960118743705774564013539, 10.60726874980758174989299347701, 11.64703361527258533819988159627, 12.72723054279008515487245429270
