| L(s) = 1 | + (−0.978 − 0.207i)2-s + (−1.44 − 2.50i)3-s + (0.913 + 0.406i)4-s + (−0.127 − 1.21i)5-s + (0.895 + 2.75i)6-s + (2.11 + 1.58i)7-s + (−0.809 − 0.587i)8-s + (−2.69 + 4.66i)9-s + (−0.127 + 1.21i)10-s + (0.615 − 5.85i)11-s + (−0.302 − 2.88i)12-s + (−1.06 − 3.26i)13-s + (−1.74 − 1.99i)14-s + (−2.86 + 2.08i)15-s + (0.669 + 0.743i)16-s + (−0.000302 + 0.00287i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.836 − 1.44i)3-s + (0.456 + 0.203i)4-s + (−0.0572 − 0.544i)5-s + (0.365 + 1.12i)6-s + (0.800 + 0.598i)7-s + (−0.286 − 0.207i)8-s + (−0.898 + 1.55i)9-s + (−0.0404 + 0.385i)10-s + (0.185 − 1.76i)11-s + (−0.0874 − 0.831i)12-s + (−0.294 − 0.906i)13-s + (−0.465 − 0.531i)14-s + (−0.741 + 0.538i)15-s + (0.167 + 0.185i)16-s + (−7.33e−5 + 0.000697i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0805159 + 0.575296i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0805159 + 0.575296i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (-2.11 - 1.58i)T \) |
| 41 | \( 1 + (6.36 + 0.716i)T \) |
| good | 3 | \( 1 + (1.44 + 2.50i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.127 + 1.21i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (-0.615 + 5.85i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (1.06 + 3.26i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.000302 - 0.00287i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (2.38 + 2.64i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (2.83 + 0.602i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (3.71 - 2.69i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.189 + 1.80i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.944 - 8.98i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (-2.39 - 7.38i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-5.27 - 1.12i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (8.83 + 3.93i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-5.94 + 6.60i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (3.35 + 3.72i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (4.36 + 1.94i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (6.07 + 4.41i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.12 + 1.94i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.17 - 2.03i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + (9.04 + 10.0i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (5.87 - 4.26i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−10.59528833508431802884263264773, −9.065047446991013929797992167742, −8.243538317503043362829285314015, −7.87318148968024361771980383894, −6.59111305660637603758424127401, −5.87620463604980647550813796742, −5.00047766816298708649912532686, −2.91159476174824688283154806931, −1.55460410588840142429784824627, −0.46699197255850440742660546263,
1.97430181183042020277125654116, 3.96962674566481595182789427983, 4.55308572811563131613310058015, 5.63774689825928782393937276783, 6.85616405936134109000355094149, 7.50917448711731210837607294299, 8.885337037787490997011256264076, 9.700885623824964763947696844075, 10.34094181626750532214245609376, 10.85788331127527792005990576542
