| L(s) = 1 | + i·3-s + i·5-s + 1.23·7-s − 9-s + 6.11·11-s − 0.591·13-s − 15-s − 4.95i·17-s + 7.62·19-s + 1.23i·21-s + (−4.72 + 0.821i)23-s − 25-s − i·27-s + 4.66·29-s − 7.50i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447i·5-s + 0.466·7-s − 0.333·9-s + 1.84·11-s − 0.164·13-s − 0.258·15-s − 1.20i·17-s + 1.74·19-s + 0.269i·21-s + (−0.985 + 0.171i)23-s − 0.200·25-s − 0.192i·27-s + 0.866·29-s − 1.34i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.479149334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.479149334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (4.72 - 0.821i)T \) |
| good | 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 13 | \( 1 + 0.591T + 13T^{2} \) |
| 17 | \( 1 + 4.95iT - 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 31 | \( 1 + 7.50iT - 31T^{2} \) |
| 37 | \( 1 - 2.86iT - 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 - 4.79T + 43T^{2} \) |
| 47 | \( 1 - 5.19iT - 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 11.5iT - 59T^{2} \) |
| 61 | \( 1 + 11.5iT - 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 71 | \( 1 + 15.7iT - 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 3.91T + 83T^{2} \) |
| 89 | \( 1 + 0.663iT - 89T^{2} \) |
| 97 | \( 1 + 11.6iT - 97T^{2} \) |
| 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
−8.057078801083859793283907218927, −7.55428642407563570777512629656, −6.65979949161419332787667606152, −6.10567685788279369457552679632, −5.16369160295483069788630232230, −4.50189160157132960248731352338, −3.69559343354551559020327251555, −3.02759718854632342294585005699, −1.91430361330510138117738358568, −0.77582384956138551952940319761,
1.13791351877730029212899168700, 1.46934993074984020614483183621, 2.71593198035152042629753087195, 3.81595424302323159562961606156, 4.32678209078180961717929835809, 5.40245566300297161515682965077, 5.98547629831343174181684128590, 6.77058281512154574257583375670, 7.38242556186632849473834531763, 8.191510119042603372926250877019
