L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.0202 + 0.0624i)3-s + (0.309 − 0.951i)4-s + (−0.536 + 2.17i)5-s + (−0.0202 − 0.0624i)6-s + 2.69·7-s + (0.309 + 0.951i)8-s + (2.42 + 1.76i)9-s + (−0.842 − 2.07i)10-s + (0.117 − 0.0853i)11-s + (0.0530 + 0.0385i)12-s + (2.21 + 1.60i)13-s + (−2.18 + 1.58i)14-s + (−0.124 − 0.0775i)15-s + (−0.809 − 0.587i)16-s + (−0.676 − 2.08i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.0117 + 0.0360i)3-s + (0.154 − 0.475i)4-s + (−0.239 + 0.970i)5-s + (−0.00827 − 0.0254i)6-s + 1.01·7-s + (0.109 + 0.336i)8-s + (0.807 + 0.586i)9-s + (−0.266 − 0.655i)10-s + (0.0354 − 0.0257i)11-s + (0.0153 + 0.0111i)12-s + (0.613 + 0.446i)13-s + (−0.583 + 0.423i)14-s + (−0.0321 − 0.0200i)15-s + (−0.202 − 0.146i)16-s + (−0.163 − 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0446 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0446 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.989812 + 0.946572i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.989812 + 0.946572i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.536 - 2.17i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (0.0202 - 0.0624i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 + (-0.117 + 0.0853i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.21 - 1.60i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.676 + 2.08i)T + (-13.7 + 9.99i)T^{2} \) |
| 23 | \( 1 + (-1.84 + 1.34i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.35 + 7.23i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.71 - 5.26i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.91 - 4.29i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.26 + 4.55i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 + (3.96 - 12.2i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.07 - 9.46i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.32 + 6.04i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-11.3 + 8.23i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.39 - 7.36i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (3.10 - 9.54i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-10.0 + 7.32i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.69 - 8.28i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.307 - 0.945i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (3.90 - 2.83i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.53 - 7.81i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.21291461874937633916882897166, −9.486858596222920976977298744497, −8.289154721484921522402543438428, −7.82454263302431944402599503005, −6.94541974458629878114264056969, −6.24518181832853787595418168744, −4.97795027521840900413109245753, −4.15171019611331661307428434259, −2.63830067699544046119454882246, −1.42509034314152519650495937352,
0.913707778124781451107956443442, 1.80710041034897657365972215477, 3.51155590755274522467262094941, 4.41739883295855320511828729217, 5.30891209667463738013566029872, 6.56664121431139432473645841404, 7.62198707561014464046124995931, 8.298265868384490140292412749585, 8.950731718852135294153436076692, 9.766475860711949599376786559599