| L(s) = 1 | + 0.732i·3-s + i·5-s − 1.26·7-s + 2.46·9-s + 3.46i·11-s + 3.46i·13-s − 0.732·15-s + 3.46·17-s + 2i·19-s − 0.928i·21-s − 8.19·23-s − 25-s + 4i·27-s + 9.46·31-s − 2.53·33-s + ⋯ |
| L(s) = 1 | + 0.422i·3-s + 0.447i·5-s − 0.479·7-s + 0.821·9-s + 1.04i·11-s + 0.960i·13-s − 0.189·15-s + 0.840·17-s + 0.458i·19-s − 0.202i·21-s − 1.70·23-s − 0.200·25-s + 0.769i·27-s + 1.69·31-s − 0.441·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00598 + 0.771917i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00598 + 0.771917i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 3 | \( 1 - 0.732iT - 3T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 10.1iT - 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4.73iT - 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−12.04821730279432438413870895065, −10.64399806852087878683562562356, −9.939274236495223674349598620674, −9.381718009093280058235558764867, −7.892735705835243592516388586619, −7.01517144691756090404615941023, −6.03801148232836937886084108644, −4.56466113048466223402066765080, −3.70419619239452920187104191926, −2.01877446154662077728623416603,
0.997678948698118200807096413095, 2.90466096968396632598197225311, 4.24128575666684809197801284385, 5.63671100692969905969956586197, 6.48637670656232606959699721908, 7.78190255295187971630639995418, 8.389806534722554268650255506042, 9.754495268399397478795114578258, 10.31407172061767210626589992281, 11.65327605309734692785898712342
