| L(s) = 1 | + (−0.0169 − 1.41i)2-s + (−0.755 + 0.654i)3-s + (−1.99 + 0.0479i)4-s + (−4.14 + 0.595i)5-s + (0.938 + 1.05i)6-s + (−0.0911 + 0.199i)7-s + (0.101 + 2.82i)8-s + (0.142 − 0.989i)9-s + (0.911 + 5.84i)10-s + (1.00 − 3.42i)11-s + (1.47 − 1.34i)12-s + (0.385 − 0.176i)13-s + (0.283 + 0.125i)14-s + (2.73 − 3.16i)15-s + (3.99 − 0.191i)16-s + (1.76 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.0119 − 0.999i)2-s + (−0.436 + 0.378i)3-s + (−0.999 + 0.0239i)4-s + (−1.85 + 0.266i)5-s + (0.383 + 0.431i)6-s + (−0.0344 + 0.0754i)7-s + (0.0359 + 0.999i)8-s + (0.0474 − 0.329i)9-s + (0.288 + 1.84i)10-s + (0.302 − 1.03i)11-s + (0.427 − 0.388i)12-s + (0.107 − 0.0488i)13-s + (0.0758 + 0.0335i)14-s + (0.707 − 0.816i)15-s + (0.998 − 0.0478i)16-s + (0.427 + 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.690722 - 0.217411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.690722 - 0.217411i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0169 + 1.41i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (-4.45 + 1.76i)T \) |
| good | 5 | \( 1 + (4.14 - 0.595i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.0911 - 0.199i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.00 + 3.42i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.385 + 0.176i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.76 - 1.13i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-3.17 - 4.94i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.283 + 0.441i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.25 - 3.75i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-3.88 - 0.558i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.393 + 2.73i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.77 + 3.27i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + (-5.83 - 2.66i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (4.28 - 1.95i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.70 - 1.47i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.22 - 4.16i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-1.36 + 0.400i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-9.58 + 6.15i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.32 - 7.28i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.57 + 0.226i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.92 + 9.14i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.773 + 5.37i)T + (-93.0 + 27.3i)T^{2} \) |
| 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
−10.97938960697383888189890209581, −10.15520799933278774107438744709, −8.929467519201612972919444474844, −8.249645980028475857250465656793, −7.33156651508190242688700352764, −5.86698105335580326386948837214, −4.71293906153312175630685490389, −3.69201451984504616885574238178, −3.21244430885066484351213133609, −0.843455261912861928008903638792,
0.73503240163827758557997926771, 3.44008425014724812269580680766, 4.52239604606167076791036962969, 5.15632823996192854093539661422, 6.66864767858209194196377309573, 7.34384484239158973934069115566, 7.82162107559022262698019495635, 8.867758134007853653871372659106, 9.738333886283679758433739774146, 11.15792882461291783736900438620
