L(s) = 1 | + (0.614 − 2.76i)2-s + (−7.24 − 3.39i)4-s + 2.97i·5-s + 7i·7-s + (−13.8 + 17.9i)8-s + (8.22 + 1.83i)10-s − 14.5·11-s + 27.9·13-s + (19.3 + 4.30i)14-s + (40.9 + 49.1i)16-s + 47.0i·17-s + 148. i·19-s + (10.1 − 21.5i)20-s + (−8.97 + 40.2i)22-s + 85.1·23-s + ⋯ |
L(s) = 1 | + (0.217 − 0.976i)2-s + (−0.905 − 0.424i)4-s + 0.266i·5-s + 0.377i·7-s + (−0.611 + 0.791i)8-s + (0.260 + 0.0579i)10-s − 0.399·11-s + 0.596·13-s + (0.368 + 0.0821i)14-s + (0.639 + 0.768i)16-s + 0.671i·17-s + 1.79i·19-s + (0.113 − 0.241i)20-s + (−0.0869 + 0.390i)22-s + 0.771·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.492920582\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.492920582\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.614 + 2.76i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 - 2.97iT - 125T^{2} \) |
| 11 | \( 1 + 14.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 148. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 85.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 197. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 213. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 3.00T + 5.06e4T^{2} \) |
| 41 | \( 1 + 4.26iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 261. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 96.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 131. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 294.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 134.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 148. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 796.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 163.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 534.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 398. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.71e3T + 9.12e5T^{2} \) |
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
−11.62123270876771712844241039905, −10.67626502327179972982466815051, −10.02289293373181787908189173613, −8.835572152395854539755076198286, −8.009120821672107551331355369274, −6.32135371725789363293671455454, −5.32717097437367149822935624189, −3.98310544011385442517665269042, −2.85655940578319420028239906052, −1.40893702155469019239209040137,
0.61024180643684794592279255481, 3.08154778882098681478647947442, 4.53228726214874839459487868121, 5.35077384978667942681411958402, 6.70856811956686364594337168702, 7.40349104462048073723857221197, 8.650888050475049324525807689857, 9.264122701603509758566525096550, 10.58666023540828582210048509105, 11.63549632358229054556423987568