L(s) = 1 | + 1.54i·5-s − 2.64·7-s + 5.28i·11-s + 17.4·13-s + 7.16i·17-s + 12.1·19-s + 41.2i·23-s + 22.6·25-s − 38.3i·29-s + 45.2·31-s − 4.08i·35-s − 52.5·37-s − 64.3i·41-s + 40.6·43-s − 79.0i·47-s + ⋯ |
L(s) = 1 | + 0.308i·5-s − 0.377·7-s + 0.480i·11-s + 1.34·13-s + 0.421i·17-s + 0.639·19-s + 1.79i·23-s + 0.904·25-s − 1.32i·29-s + 1.45·31-s − 0.116i·35-s − 1.42·37-s − 1.56i·41-s + 0.946·43-s − 1.68i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.259852617\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.259852617\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 - 1.54iT - 25T^{2} \) |
| 11 | \( 1 - 5.28iT - 121T^{2} \) |
| 13 | \( 1 - 17.4T + 169T^{2} \) |
| 17 | \( 1 - 7.16iT - 289T^{2} \) |
| 19 | \( 1 - 12.1T + 361T^{2} \) |
| 23 | \( 1 - 41.2iT - 529T^{2} \) |
| 29 | \( 1 + 38.3iT - 841T^{2} \) |
| 31 | \( 1 - 45.2T + 961T^{2} \) |
| 37 | \( 1 + 52.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 64.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 79.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 88.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 39.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 13.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 62.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 12.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 114.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 42.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 9.41iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 63.2T + 9.40e3T^{2} \) |
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
−8.497307968726331192078941428121, −7.48409671368162571530925838674, −7.04947953048851475692057461338, −6.04055599634246699074409248707, −5.65440672148237477699476846901, −4.51426162896856051422091476235, −3.69375263974406066324744587585, −3.04093916476978292948363105285, −1.88958218071960813009789455760, −0.897309340814179025285925814481,
0.61187404405621629515749140144, 1.41233775932342088911349448128, 2.85761402678083986719336260638, 3.34347318236444108129038441700, 4.48806471902949407207424482213, 5.05983454104817179318994196653, 6.18372935268611906798444377793, 6.47396038033276588263564522840, 7.42855461859039836492812817537, 8.450523120284837992481466145954