| L(s) = 1 | − 5.48i·7-s + 9.17i·11-s − 11.4i·13-s + 16.9·17-s + 26.9·19-s + 4.93·23-s − 20.5i·29-s − 20.9·31-s − 62.4i·37-s + 40.9i·41-s + 1.02i·43-s − 86.2·47-s + 18.8·49-s + 96.0·53-s + 112. i·59-s + ⋯ |
| L(s) = 1 | − 0.783i·7-s + 0.833i·11-s − 0.883i·13-s + 0.998·17-s + 1.41·19-s + 0.214·23-s − 0.707i·29-s − 0.676·31-s − 1.68i·37-s + 0.998i·41-s + 0.0238i·43-s − 1.83·47-s + 0.385·49-s + 1.81·53-s + 1.90i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.153330674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153330674\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5.48iT - 49T^{2} \) |
| 11 | \( 1 - 9.17iT - 121T^{2} \) |
| 13 | \( 1 + 11.4iT - 169T^{2} \) |
| 17 | \( 1 - 16.9T + 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 - 4.93T + 529T^{2} \) |
| 29 | \( 1 + 20.5iT - 841T^{2} \) |
| 31 | \( 1 + 20.9T + 961T^{2} \) |
| 37 | \( 1 + 62.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 40.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.02iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 86.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 96.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 24.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 18.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 106.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 45.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.0iT - 9.40e3T^{2} \) |
| 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
−8.020118002499470682909264604744, −7.47859879254787180355962084187, −7.05117202533504485222664571395, −5.85069068316789812185985222902, −5.31762318555143393082700106652, −4.37388022650530652310278744729, −3.56585204445712796939871843692, −2.74996598734631606075461287573, −1.48444606226197784204272505015, −0.56671188613888617600924529659,
0.944940200521089733427858468292, 1.95780524395935948719592033846, 3.12600441872528749601359980843, 3.60675801804261926995363469033, 4.96745882917456049043078104438, 5.39735456158546838408875489119, 6.26097160999279850678484653841, 6.98880166854138419755298705781, 7.86204354146352740098340216646, 8.506315218846377589883779275963
