L(s) = 1 | + (−0.987 + 0.156i)2-s + (0.487 + 1.66i)3-s + (0.951 − 0.309i)4-s + (2.22 + 0.193i)5-s + (−0.741 − 1.56i)6-s + (−0.104 − 0.104i)7-s + (−0.891 + 0.453i)8-s + (−2.52 + 1.62i)9-s + (−2.23 + 0.157i)10-s + (1.81 − 2.49i)11-s + (0.977 + 1.42i)12-s + (−0.711 + 4.49i)13-s + (0.119 + 0.0869i)14-s + (0.765 + 3.79i)15-s + (0.809 − 0.587i)16-s + (1.68 + 3.30i)17-s + ⋯ |
L(s) = 1 | + (−0.698 + 0.110i)2-s + (0.281 + 0.959i)3-s + (0.475 − 0.154i)4-s + (0.996 + 0.0863i)5-s + (−0.302 − 0.638i)6-s + (−0.0395 − 0.0395i)7-s + (−0.315 + 0.160i)8-s + (−0.841 + 0.540i)9-s + (−0.705 + 0.0499i)10-s + (0.546 − 0.752i)11-s + (0.282 + 0.412i)12-s + (−0.197 + 1.24i)13-s + (0.0319 + 0.0232i)14-s + (0.197 + 0.980i)15-s + (0.202 − 0.146i)16-s + (0.408 + 0.801i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.862968 + 0.521092i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.862968 + 0.521092i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.987 - 0.156i)T \) |
| 3 | \( 1 + (-0.487 - 1.66i)T \) |
| 5 | \( 1 + (-2.22 - 0.193i)T \) |
good | 7 | \( 1 + (0.104 + 0.104i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.81 + 2.49i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.711 - 4.49i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.68 - 3.30i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (3.32 + 1.08i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.773 + 4.88i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.509 + 1.56i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.45 + 7.56i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (8.30 + 1.31i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (1.27 + 1.75i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.77 + 2.77i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.77 - 2.43i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-5.24 + 10.2i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (10.4 - 7.57i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-11.8 - 8.58i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.23 + 1.64i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (4.79 - 1.55i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (10.6 - 1.68i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-8.57 + 2.78i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (9.01 - 4.59i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-6.13 - 4.45i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.20 - 8.26i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−13.46915881438020372275230545055, −11.89126224466003741982749524248, −10.78659299690367398630304297874, −10.06105618217741976638911495651, −9.091559558134752475793080783465, −8.464756265917050090055219144202, −6.69519132013679640796088341502, −5.69022772548555099244200485775, −4.06294334088641262266437646800, −2.26722932499397721999701850583,
1.50803087394075448672505676506, 2.93945904682430027384171912671, 5.44112315725967465731956123865, 6.63054683728749249068120754977, 7.58839816401912071805948503485, 8.746888336719848188525350723543, 9.655674530011888972267383720697, 10.63352957398576593280002286501, 12.07986629932997149314752500304, 12.68237666640749392027992911240