| L(s) = 1 | + (0.712 + 1.22i)2-s + (−0.243 + 0.124i)3-s + (−0.983 + 1.74i)4-s + (0.914 − 2.04i)5-s + (−0.325 − 0.209i)6-s + (3.34 + 3.34i)7-s + (−2.82 + 0.0403i)8-s + (−1.71 + 2.36i)9-s + (3.14 − 0.337i)10-s + (−0.383 + 0.278i)11-s + (0.0234 − 0.546i)12-s + (2.92 − 0.463i)13-s + (−1.69 + 6.46i)14-s + (0.0304 + 0.610i)15-s + (−2.06 − 3.42i)16-s + (−5.33 − 2.71i)17-s + ⋯ |
| L(s) = 1 | + (0.504 + 0.863i)2-s + (−0.140 + 0.0716i)3-s + (−0.491 + 0.870i)4-s + (0.409 − 0.912i)5-s + (−0.132 − 0.0853i)6-s + (1.26 + 1.26i)7-s + (−0.999 + 0.0142i)8-s + (−0.573 + 0.788i)9-s + (0.994 − 0.106i)10-s + (−0.115 + 0.0839i)11-s + (0.00676 − 0.157i)12-s + (0.810 − 0.128i)13-s + (−0.454 + 1.72i)14-s + (0.00785 + 0.157i)15-s + (−0.516 − 0.856i)16-s + (−1.29 − 0.659i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0746 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0746 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12291 + 1.04203i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12291 + 1.04203i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.712 - 1.22i)T \) |
| 5 | \( 1 + (-0.914 + 2.04i)T \) |
| good | 3 | \( 1 + (0.243 - 0.124i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (-3.34 - 3.34i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.383 - 0.278i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.92 + 0.463i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (5.33 + 2.71i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-3.39 + 1.10i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.486 + 3.06i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (0.303 - 0.933i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.47 + 1.12i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.28 + 8.12i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-1.86 - 1.35i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-1.60 - 1.60i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.14 - 1.09i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (4.51 + 8.87i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (4.70 - 6.46i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.48 + 2.04i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (11.2 + 5.71i)T + (39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-6.38 - 2.07i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.46 - 1.02i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (4.09 - 12.5i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.82 + 7.49i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (6.17 + 8.50i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.79 + 5.49i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−12.86789128251864587428786357722, −11.83273719553727054453773503138, −11.10185211962102797979148339830, −9.174879954252656688382445924392, −8.606706123817987863140974419203, −7.81233407051197517048357859622, −6.12607003441038630469783201445, −5.22667663544523321282387681146, −4.64631404544194382893180511644, −2.39781782863383906194825531178,
1.50339719632251276335996244715, 3.27845934530507481903322284124, 4.39818131375970992197722979672, 5.82953842814293102221175425256, 6.84936438658584689854938255670, 8.312861067900929421486507724801, 9.591851018572016035941671924457, 10.77567651013480360555008135780, 11.06834498866919464106516363601, 11.96374564093996658611480889049
