| L(s) = 1 | + (−1 + i)3-s − 2i·7-s + i·9-s + (−1 − i)11-s + (1 − i)13-s + 2·17-s + (−3 + 3i)19-s + (2 + 2i)21-s + 6i·23-s + (−4 − 4i)27-s + (3 − 3i)29-s + 8·31-s + 2·33-s + (−3 − 3i)37-s + 2i·39-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s − 0.755i·7-s + 0.333i·9-s + (−0.301 − 0.301i)11-s + (0.277 − 0.277i)13-s + 0.485·17-s + (−0.688 + 0.688i)19-s + (0.436 + 0.436i)21-s + 1.25i·23-s + (−0.769 − 0.769i)27-s + (0.557 − 0.557i)29-s + 1.43·31-s + 0.348·33-s + (−0.493 − 0.493i)37-s + 0.320i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216761630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216761630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + (1 + i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + (3 - 3i)T - 19iT^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-3 + 3i)T - 29iT^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-5 - 5i)T + 43iT^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3 - 3i)T + 59iT^{2} \) |
| 61 | \( 1 + (9 - 9i)T - 61iT^{2} \) |
| 67 | \( 1 + (5 - 5i)T - 67iT^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (1 - i)T - 83iT^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.872932570322977685621864201960, −8.741725797543948219256329406089, −7.895540715125077742601543027210, −7.28393738337688952010220885128, −6.03780032195121777799909554856, −5.56488020806183165662227679280, −4.47398920521395097767225012078, −3.85977087013884148245362238782, −2.61337118097625187623038090185, −1.05445676113746192266653822891,
0.62372294225937580169693360239, 2.03026736438775170760413857794, 3.05563059111024567756532877868, 4.36555707860134889009954967928, 5.23603551650315258810324566151, 6.20228723236467261597295882552, 6.63568898522911555025553376384, 7.56250096929534773600998650581, 8.599073138777116856074738628014, 9.055087284914310541743538158781
