L(s) = 1 | − 3-s − 2.37·5-s + 2.52i·7-s + 9-s − 2.52i·11-s + 1.58i·13-s + 2.37·15-s − 0.372·17-s + (−4 − 1.73i)19-s − 2.52i·21-s + 1.87i·23-s + 0.627·25-s − 27-s − 3.16i·29-s − 2.74·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.06·5-s + 0.954i·7-s + 0.333·9-s − 0.761i·11-s + 0.439i·13-s + 0.612·15-s − 0.0902·17-s + (−0.917 − 0.397i)19-s − 0.550i·21-s + 0.391i·23-s + 0.125·25-s − 0.192·27-s − 0.588i·29-s − 0.492·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7698624600\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7698624600\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2.52iT - 7T^{2} \) |
| 11 | \( 1 + 2.52iT - 11T^{2} \) |
| 13 | \( 1 - 1.58iT - 13T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 23 | \( 1 - 1.87iT - 23T^{2} \) |
| 29 | \( 1 + 3.16iT - 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 1.58iT - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 0.644iT - 43T^{2} \) |
| 47 | \( 1 - 0.939iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 13.2iT - 89T^{2} \) |
| 97 | \( 1 - 13.2iT - 97T^{2} \) |
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
−8.603140654149090439289112456222, −7.72028514704135591367180865372, −7.05026864304093761782166540783, −6.11307901640061332796790238905, −5.63496159089670454858487777701, −4.60777882992524230117807493876, −3.96764876536958267130292578021, −3.00108542774397664678244841986, −1.92152701358001627327760603063, −0.41669240776022289053565060875,
0.64346358949394989925940538632, 1.92060744259310102309015794400, 3.31192593420858052323894264772, 4.11401816176568410977028216705, 4.58584482033290902410095045860, 5.53138354181609709283848671363, 6.55765202708998235270720168978, 7.10994681736966551096855477590, 7.78046722425313772484725694017, 8.358899835122256264506428274641