| L(s) = 1 | + (0.809 + 0.587i)2-s + (0.104 + 0.994i)3-s + (0.309 + 0.951i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.744 + 0.158i)7-s + (−0.309 + 0.951i)8-s + (−0.978 + 0.207i)9-s + (0.104 − 0.994i)10-s + (3.69 − 4.10i)11-s + (−0.913 + 0.406i)12-s + (2.55 + 1.13i)13-s + (0.508 + 0.565i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (4.97 + 5.52i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.0603 + 0.574i)3-s + (0.154 + 0.475i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (0.281 + 0.0597i)7-s + (−0.109 + 0.336i)8-s + (−0.326 + 0.0693i)9-s + (0.0330 − 0.314i)10-s + (1.11 − 1.23i)11-s + (−0.263 + 0.117i)12-s + (0.708 + 0.315i)13-s + (0.136 + 0.151i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (1.20 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94393 + 1.36970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94393 + 1.36970i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-3.60 - 4.24i)T \) |
| good | 7 | \( 1 + (-0.744 - 0.158i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-3.69 + 4.10i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.55 - 1.13i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-4.97 - 5.52i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.699 - 0.311i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.936 - 2.88i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.44 + 1.77i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (2.95 - 5.12i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.266 + 2.53i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (3.68 - 1.64i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-4.33 + 3.15i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.893 + 0.189i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.170 - 1.61i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + (4.83 + 8.37i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.8 + 2.73i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-8.80 + 9.78i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-0.935 - 1.03i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.169 - 1.61i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.755 - 2.32i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.47 + 10.7i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.29363761521146087179175512776, −9.186358362701015512645623972880, −8.463005416743115221594670421086, −7.927475497528310078512266539314, −6.50101763502991093974398311016, −5.89851123922975967189884287518, −4.97586243216883836695203103921, −3.78618627280099471916482282492, −3.46862933223974849416469906141, −1.45546615108778070634894601740,
1.13238610861481153496452900851, 2.35498950574098912109033237742, 3.49223462498828352126022565817, 4.44606128255372135005331876133, 5.51000622166877305368727721371, 6.54400979672945819595093900218, 7.21500661729847808834580441381, 8.059361823972356079070915473357, 9.250815336835449420149333349077, 9.943438423397745878379751782952
