| L(s) = 1 | − 3.80i·3-s − 8.50i·7-s − 5.47·9-s + 1.79i·11-s + 0.472·13-s + 23.8·17-s − 9.40i·19-s − 32.3·21-s − 16.1i·23-s − 13.4i·27-s − 6.94·29-s − 47.4i·31-s + 6.83·33-s + 26.3·37-s − 1.79i·39-s + ⋯ |
| L(s) = 1 | − 1.26i·3-s − 1.21i·7-s − 0.608·9-s + 0.163i·11-s + 0.0363·13-s + 1.40·17-s − 0.494i·19-s − 1.54·21-s − 0.700i·23-s − 0.497i·27-s − 0.239·29-s − 1.53i·31-s + 0.207·33-s + 0.712·37-s − 0.0460i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.714477428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714477428\) |
| \(L(2)\) |
|
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 | \( 1 + 3.80iT - 9T^{2} \) |
| 7 | \( 1 + 8.50iT - 49T^{2} \) |
| 11 | \( 1 - 1.79iT - 121T^{2} \) |
| 13 | \( 1 - 0.472T + 169T^{2} \) |
| 17 | \( 1 - 23.8T + 289T^{2} \) |
| 19 | \( 1 + 9.40iT - 361T^{2} \) |
| 23 | \( 1 + 16.1iT - 529T^{2} \) |
| 29 | \( 1 + 6.94T + 841T^{2} \) |
| 31 | \( 1 + 47.4iT - 961T^{2} \) |
| 37 | \( 1 - 26.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 2.00iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 21.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 21.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 39.1T + 9.40e3T^{2} \) |
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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
−8.578727946851132888461303767201, −7.70780625033564457019462622652, −7.36722913312891983569337414795, −6.58172625495054262298727144286, −5.77644408756162896115930569700, −4.59306253249796545704344220767, −3.65768636003878801359248200622, −2.45999724924237791217483906537, −1.30530997433191650352184705718, −0.49620220960848057048695946021,
1.54540582187602917110336099787, 2.99734317827337944769037414648, 3.59370712864103491465834633755, 4.75243686139862547226981299921, 5.43257647416462777831871753991, 6.05139988342334299685855318886, 7.31363991503926746316450064470, 8.286769328959605906941240467654, 8.952950585509215799094874626151, 9.683794247364254840215408309613
