| L(s) = 1 | + 0.732i·3-s + 2i·7-s + 2.46·9-s + 3.46·11-s − 0.732i·13-s + 3.46i·17-s − 19-s − 1.46·21-s − 3.46i·23-s + 4i·27-s + 3.46·29-s + 5.46·31-s + 2.53i·33-s + 3.26i·37-s + 0.535·39-s + ⋯ |
| L(s) = 1 | + 0.422i·3-s + 0.755i·7-s + 0.821·9-s + 1.04·11-s − 0.203i·13-s + 0.840i·17-s − 0.229·19-s − 0.319·21-s − 0.722i·23-s + 0.769i·27-s + 0.643·29-s + 0.981·31-s + 0.441i·33-s + 0.537i·37-s + 0.0858·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.988000364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.988000364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.732iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 0.732iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 - 3.26iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.92iT - 43T^{2} \) |
| 47 | \( 1 + 0.928iT - 47T^{2} \) |
| 53 | \( 1 - 7.26iT - 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 - 3.26iT - 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 7.46iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 8.53T + 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
| 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
−9.224231267302350229046190615070, −8.742545113006784224603516813533, −7.919797756248267201160876603906, −6.79266269582320876917796880686, −6.28647529078127390787772412907, −5.24693946134671104632165963438, −4.36732104761944392557927208307, −3.65443757765760479135605192137, −2.43818718212792993491526614566, −1.26417365099105297250291928728,
0.860208145605935195256676104336, 1.81405962484030037348432388826, 3.20248208932546301336539203707, 4.18555733333832005450242237785, 4.81459529214906103461230816430, 6.13489504513085329214374585161, 6.81719956167139161890074942880, 7.36285729255168504995586090737, 8.179167796339629085011106818586, 9.171513723042375464678996610501
