L(s) = 1 | + (−2.84 − 0.938i)3-s + 6.81·7-s + (7.23 + 5.34i)9-s + 7.52i·11-s − 16.2·13-s − 4.11i·17-s − 7.86·19-s + (−19.4 − 6.39i)21-s − 19.5i·23-s + (−15.6 − 22.0i)27-s + 55.8i·29-s + 43.4·31-s + (7.06 − 21.4i)33-s − 31.5·37-s + (46.2 + 15.2i)39-s + ⋯ |
L(s) = 1 | + (−0.949 − 0.312i)3-s + 0.973·7-s + (0.804 + 0.594i)9-s + 0.684i·11-s − 1.24·13-s − 0.242i·17-s − 0.413·19-s + (−0.924 − 0.304i)21-s − 0.848i·23-s + (−0.578 − 0.815i)27-s + 1.92i·29-s + 1.40·31-s + (0.214 − 0.650i)33-s − 0.853·37-s + (1.18 + 0.390i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.034952638\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.034952638\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.84 + 0.938i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 6.81T + 49T^{2} \) |
| 11 | \( 1 - 7.52iT - 121T^{2} \) |
| 13 | \( 1 + 16.2T + 169T^{2} \) |
| 17 | \( 1 + 4.11iT - 289T^{2} \) |
| 19 | \( 1 + 7.86T + 361T^{2} \) |
| 23 | \( 1 + 19.5iT - 529T^{2} \) |
| 29 | \( 1 - 55.8iT - 841T^{2} \) |
| 31 | \( 1 - 43.4T + 961T^{2} \) |
| 37 | \( 1 + 31.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 61.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 82.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 97.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 4.13T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 78.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.72iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 70.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 3.44T + 9.40e3T^{2} \) |
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
−10.67029720228699956048231709747, −10.03470085641950757274650762842, −8.866809768137054366881429753362, −7.72040857444457193324481445056, −7.12590019018097502074422621056, −6.11015188571404323311469711827, −4.84043954642962917070198591788, −4.60860533745454495109270812438, −2.52829389847606497412828907473, −1.26328109814968183858291089520,
0.48007488244534129650297865342, 2.09919146711306146099553454641, 3.82703929744634877174261275878, 4.83120313699214708430639410917, 5.55135164987865776634010694075, 6.57165129149733740058200324299, 7.60224068921220156314334019981, 8.471000890311816504454671785057, 9.678286324866104341303565551785, 10.31276717794054900638611904004