| L(s) = 1 | + (0.0977 − 1.11i)2-s + (2.70 + 0.476i)4-s + (0.528 − 4.97i)5-s + (2.01 + 1.41i)7-s + (1.95 − 7.30i)8-s + (−5.50 − 1.07i)10-s + (−13.4 − 4.88i)11-s + (−0.975 − 11.1i)13-s + (1.77 − 2.11i)14-s + (2.34 + 0.851i)16-s + (0.0303 + 0.113i)17-s + (7.83 − 4.52i)19-s + (3.79 − 13.1i)20-s + (−6.77 + 14.5i)22-s + (8.56 − 6.00i)23-s + ⋯ |
| L(s) = 1 | + (0.0488 − 0.558i)2-s + (0.675 + 0.119i)4-s + (0.105 − 0.994i)5-s + (0.288 + 0.201i)7-s + (0.244 − 0.912i)8-s + (−0.550 − 0.107i)10-s + (−1.22 − 0.444i)11-s + (−0.0750 − 0.857i)13-s + (0.126 − 0.151i)14-s + (0.146 + 0.0532i)16-s + (0.00178 + 0.00666i)17-s + (0.412 − 0.238i)19-s + (0.189 − 0.658i)20-s + (−0.307 + 0.660i)22-s + (0.372 − 0.260i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.00339 - 1.68892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00339 - 1.68892i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.528 + 4.97i)T \) |
| good | 2 | \( 1 + (-0.0977 + 1.11i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-2.01 - 1.41i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (13.4 + 4.88i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.975 + 11.1i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-0.0303 - 0.113i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-7.83 + 4.52i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-8.56 + 6.00i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-21.1 - 25.1i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (0.497 - 2.82i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-2.58 - 9.66i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (39.9 + 33.5i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (33.8 + 72.5i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-59.0 - 41.3i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-45.2 - 45.2i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (17.7 + 48.7i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-6.33 - 35.9i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-18.4 + 1.61i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-21.1 + 36.5i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (25.1 - 93.9i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (7.64 + 9.11i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-101. - 8.84i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (126. - 73.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-147. + 68.8i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.67250955624515521533823140485, −10.16554992500955770381209903610, −8.849780673253138550467504105919, −8.073547128549066887864955783217, −7.10346493052611767071478397402, −5.69225539090158613283811940790, −4.94742708911044567258299965748, −3.41017851608367163210701132883, −2.28883265304458840628126976609, −0.817191787116971369157781868381,
1.97013856976572569187160502921, 3.01816160255063809741323465432, 4.68356628053793651863564961886, 5.80159460624353467315910951082, 6.74522652330769147021954370017, 7.45997922374005282085918016480, 8.197818399028743910537223012286, 9.753723016227910486911006068823, 10.46768536077522120969201900178, 11.29357725016003346398016968396
