| L(s) = 1 | − 1.83i·2-s − 1.36·4-s + 3.80·5-s − 1.16i·8-s − 6.98i·10-s + 1.16i·11-s − 5.54i·13-s − 4.86·16-s − 3.17·17-s + 0.631i·19-s − 5.19·20-s + 2.13·22-s + 1.76i·23-s + 9.50·25-s − 10.1·26-s + ⋯ |
| L(s) = 1 | − 1.29i·2-s − 0.682·4-s + 1.70·5-s − 0.412i·8-s − 2.20i·10-s + 0.351i·11-s − 1.53i·13-s − 1.21·16-s − 0.770·17-s + 0.144i·19-s − 1.16·20-s + 0.455·22-s + 0.367i·23-s + 1.90·25-s − 1.99·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237734705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237734705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.83iT - 2T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 11 | \( 1 - 1.16iT - 11T^{2} \) |
| 13 | \( 1 + 5.54iT - 13T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 19 | \( 1 - 0.631iT - 19T^{2} \) |
| 23 | \( 1 - 1.76iT - 23T^{2} \) |
| 29 | \( 1 + 4.83iT - 29T^{2} \) |
| 31 | \( 1 + 4.27iT - 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 + 4.56T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 - 0.0724iT - 53T^{2} \) |
| 59 | \( 1 - 6.98T + 59T^{2} \) |
| 61 | \( 1 + 8.08iT - 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 - 4.76iT - 71T^{2} \) |
| 73 | \( 1 - 5.59iT - 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 0.688iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.708980543532541127457833935808, −8.952337484452610383597414382872, −7.81861886299112426922219712856, −6.65876646497709966703490792121, −5.88639500562803303649008150851, −5.05530167084641579061582530400, −3.86301254059275621503379818004, −2.65030383181059154347683321725, −2.15610915989791104181754962439, −0.934795940253944109852513375695,
1.70855010363566302671780848008, 2.61452226453338698656337230744, 4.36896309234603504337592605332, 5.23051344352144488835563741876, 5.97285653255529635637238431961, 6.66788759040339175291874035353, 7.08172527582846547787978568363, 8.446686605123535551425038440022, 9.008026742839384467966436103483, 9.572944999474158315747552491620
