| L(s) = 1 | + (−1.24 + 0.664i)2-s + (1.66 + 0.464i)3-s + (1.11 − 1.65i)4-s + (0.345 − 2.20i)5-s + (−2.39 + 0.528i)6-s − 0.309i·7-s + (−0.292 + 2.81i)8-s + (2.56 + 1.55i)9-s + (1.03 + 2.98i)10-s + (1.53 − 1.11i)11-s + (2.63 − 2.24i)12-s + (−3.79 − 2.75i)13-s + (0.205 + 0.386i)14-s + (1.60 − 3.52i)15-s + (−1.50 − 3.70i)16-s + (4.30 − 1.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.882 + 0.469i)2-s + (0.963 + 0.268i)3-s + (0.558 − 0.829i)4-s + (0.154 − 0.987i)5-s + (−0.976 + 0.215i)6-s − 0.117i·7-s + (−0.103 + 0.994i)8-s + (0.856 + 0.516i)9-s + (0.327 + 0.944i)10-s + (0.463 − 0.336i)11-s + (0.760 − 0.649i)12-s + (−1.05 − 0.764i)13-s + (0.0550 + 0.103i)14-s + (0.413 − 0.910i)15-s + (−0.376 − 0.926i)16-s + (1.04 − 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24318 - 0.0662395i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24318 - 0.0662395i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.24 - 0.664i)T \) |
| 3 | \( 1 + (-1.66 - 0.464i)T \) |
| 5 | \( 1 + (-0.345 + 2.20i)T \) |
| good | 7 | \( 1 + 0.309iT - 7T^{2} \) |
| 11 | \( 1 + (-1.53 + 1.11i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.79 + 2.75i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.30 + 1.39i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.24 + 1.70i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (5.20 - 3.78i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.49 - 2.11i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (5.73 - 1.86i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.24 - 2.35i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.902 + 1.24i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.75iT - 43T^{2} \) |
| 47 | \( 1 + (2.19 - 6.75i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.721 + 0.234i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (11.0 + 8.03i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.66 - 5.57i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (6.37 - 2.07i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.750 - 2.30i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (7.60 - 5.52i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.81 - 0.588i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.792 - 2.43i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.28 + 7.27i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.88 + 11.9i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.68879110592187035193188447205, −10.23890027437950335308041322900, −9.637917805244703916857927098786, −8.963487811643040246284418814777, −7.906328834711951258180343341613, −7.41728395864740966659000935533, −5.74291181082622518298112122708, −4.76649265387799289167751196516, −3.01388481081487460293901218514, −1.30625589269553154722417135374,
1.84950474933916891494348332658, 2.90425375109737807215220957440, 4.02315380718052143873522956192, 6.28939925257397372450231537057, 7.32850911580834161870555366559, 7.83291207936640277215621245015, 9.135298448726996430542600455455, 9.827304296779867365672812902417, 10.46244064965271648703490967702, 12.00294709411968856827033952413
