| L(s) = 1 | + 4.94·3-s − 2.23·5-s + 13.1i·7-s + 15.4·9-s + (3.18 − 10.5i)11-s + 14.6i·13-s − 11.0·15-s + 9.90i·17-s − 11.5i·19-s + 65.1i·21-s + 14.6·23-s + 5.00·25-s + 32.1·27-s + 8.28i·29-s − 53.2·31-s + ⋯ |
| L(s) = 1 | + 1.64·3-s − 0.447·5-s + 1.87i·7-s + 1.72·9-s + (0.289 − 0.957i)11-s + 1.12i·13-s − 0.737·15-s + 0.582i·17-s − 0.607i·19-s + 3.10i·21-s + 0.634·23-s + 0.200·25-s + 1.18·27-s + 0.285i·29-s − 1.71·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.065540730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.065540730\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 11 | \( 1 + (-3.18 + 10.5i)T \) |
| good | 3 | \( 1 - 4.94T + 9T^{2} \) |
| 7 | \( 1 - 13.1iT - 49T^{2} \) |
| 13 | \( 1 - 14.6iT - 169T^{2} \) |
| 17 | \( 1 - 9.90iT - 289T^{2} \) |
| 19 | \( 1 + 11.5iT - 361T^{2} \) |
| 23 | \( 1 - 14.6T + 529T^{2} \) |
| 29 | \( 1 - 8.28iT - 841T^{2} \) |
| 31 | \( 1 + 53.2T + 961T^{2} \) |
| 37 | \( 1 - 41.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 74.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 30.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 0.697T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 6.29T + 3.48e3T^{2} \) |
| 61 | \( 1 - 50.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 81.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 62.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 11.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 79.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 81.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 136.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 80.2T + 9.40e3T^{2} \) |
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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
−9.548528721756706227864531011271, −9.101525998778302450947466210941, −8.589884251985388009400182381450, −7.936718033666346898477710281774, −6.80247481047988120138601656122, −5.81553700585058701682351318406, −4.55973563085333391043596273243, −3.43202027956531526655779811604, −2.70768976127991422738887316268, −1.72899067361734861712004121724,
0.833158018267624353636657838134, 2.21843980295417926625709074006, 3.60661710259071283734536449613, 3.85130642970399287130672488408, 5.03254871417386596785604721164, 6.85059186254765780715257519487, 7.54285448221378025608042302867, 7.80548490447029494997303773786, 8.919627726614641842113254679695, 9.750184013989932809289324789513
