| L(s) = 1 | + 8.92·5-s − 10.9i·7-s + (1.19 − 10.9i)11-s + 8.48i·13-s − 20.2i·17-s + 6.03i·19-s − 18.3·23-s + 54.7·25-s − 38.5i·29-s + 14.1·31-s − 97.6i·35-s − 31.7·37-s − 13.6i·41-s + 30.8i·43-s + 12.7·47-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.56i·7-s + (0.108 − 0.994i)11-s + 0.652i·13-s − 1.19i·17-s + 0.317i·19-s − 0.799·23-s + 2.18·25-s − 1.32i·29-s + 0.456·31-s − 2.78i·35-s − 0.857·37-s − 0.333i·41-s + 0.718i·43-s + 0.272·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.626548520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626548520\) |
| \(L(2)\) |
|
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 | \( 1 + (-1.19 + 10.9i)T \) |
| good | 5 | \( 1 - 8.92T + 25T^{2} \) |
| 7 | \( 1 + 10.9iT - 49T^{2} \) |
| 13 | \( 1 - 8.48iT - 169T^{2} \) |
| 17 | \( 1 + 20.2iT - 289T^{2} \) |
| 19 | \( 1 - 6.03iT - 361T^{2} \) |
| 23 | \( 1 + 18.3T + 529T^{2} \) |
| 29 | \( 1 + 38.5iT - 841T^{2} \) |
| 31 | \( 1 - 14.1T + 961T^{2} \) |
| 37 | \( 1 + 31.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 30.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 12.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 25.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 35.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 7.17iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 98.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 94.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 4.11iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 43.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 50T + 7.92e3T^{2} \) |
| 97 | \( 1 - 34.7T + 9.40e3T^{2} \) |
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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
−9.283535444715975206534275256369, −8.260627637604674473897561751204, −7.27999143731516531638237621577, −6.49177861734777695026841091617, −5.90006888497028868901541036259, −4.91109764064157434916104234932, −3.96487813210149598857315018704, −2.81136982429882960336159632739, −1.69483852204590772852321802614, −0.66919619303953489295861720517,
1.65160877785736323418041072955, 2.17087512061758748756079176408, 3.16795045632235101228349732864, 4.76942857488870364131771690728, 5.49557750019512032316951125292, 6.06037113374844597751648581902, 6.75717434390909519437232813539, 8.020779723567674415360179168714, 8.949460918101930185430079866154, 9.293551719494431787784156452064
