| L(s) = 1 | + 6.78·5-s − 11.7i·7-s + (6.78 + 8.66i)11-s − 11.7i·13-s + 10.3i·17-s + 33.9·23-s + 21·25-s + 34.6i·29-s + 10·31-s − 79.6i·35-s + 50·37-s − 34.6i·41-s − 46.9i·43-s + 33.9·47-s − 89·49-s + ⋯ |
| L(s) = 1 | + 1.35·5-s − 1.67i·7-s + (0.616 + 0.787i)11-s − 0.903i·13-s + 0.611i·17-s + 1.47·23-s + 0.839·25-s + 1.19i·29-s + 0.322·31-s − 2.27i·35-s + 1.35·37-s − 0.844i·41-s − 1.09i·43-s + 0.721·47-s − 1.81·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.871328612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.871328612\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-6.78 - 8.66i)T \) |
| good | 5 | \( 1 - 6.78T + 25T^{2} \) |
| 7 | \( 1 + 11.7iT - 49T^{2} \) |
| 13 | \( 1 + 11.7iT - 169T^{2} \) |
| 17 | \( 1 - 10.3iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 33.9T + 529T^{2} \) |
| 29 | \( 1 - 34.6iT - 841T^{2} \) |
| 31 | \( 1 - 10T + 961T^{2} \) |
| 37 | \( 1 - 50T + 1.36e3T^{2} \) |
| 41 | \( 1 + 34.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 46.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 33.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 33.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 67.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 58.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 10T + 4.48e3T^{2} \) |
| 71 | \( 1 - 33.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 70.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 58.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 13.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 40T + 9.40e3T^{2} \) |
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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
−9.384405075008442469118646509301, −8.342079494557090916253949219942, −7.26811773482576794041860615474, −6.84163817278684635755301681377, −5.88332131223994740233773845387, −4.97974207891604216452892438901, −4.09068231431151822728766965703, −3.04810176919070320144342292654, −1.72725396147049061213118600968, −0.873291499035421169695954093839,
1.24011989332734649831187638607, 2.35210011089353311597843892014, 2.96081044512829067150874946444, 4.53133390531323746155783881711, 5.41044168329973975049774698946, 6.15940414870605225786838173864, 6.51390099071216163798739813744, 7.895152349587945571485411279228, 8.922966563979174928484135581098, 9.273428431849614292083181486924
