| L(s) = 1 | − 1.41·2-s + (1.41 − 2.64i)3-s + 2.00·4-s + (−2.00 + 3.74i)6-s − 2.64i·7-s − 2.82·8-s + (−5 − 7.48i)9-s − 14.5i·11-s + (2.82 − 5.29i)12-s + 0.583i·13-s + 3.74i·14-s + 4.00·16-s + 21.5·17-s + (7.07 + 10.5i)18-s + 16·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.471 − 0.881i)3-s + 0.500·4-s + (−0.333 + 0.623i)6-s − 0.377i·7-s − 0.353·8-s + (−0.555 − 0.831i)9-s − 1.32i·11-s + (0.235 − 0.440i)12-s + 0.0448i·13-s + 0.267i·14-s + 0.250·16-s + 1.26·17-s + (0.392 + 0.587i)18-s + 0.842·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.430775275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.430775275\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (-1.41 + 2.64i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 + 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 0.583iT - 169T^{2} \) |
| 17 | \( 1 - 21.5T + 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 - 38.8T + 529T^{2} \) |
| 29 | \( 1 + 35.7iT - 841T^{2} \) |
| 31 | \( 1 + 58.4T + 961T^{2} \) |
| 37 | \( 1 - 20iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.75iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 8.48T + 2.20e3T^{2} \) |
| 53 | \( 1 + 50.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 70.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 72.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 145.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 53.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 111. iT - 9.40e3T^{2} \) |
| 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
−9.224777102598392007604880994879, −8.525226261769691608436180255663, −7.71119392674272912578029295076, −7.18503824371858710738429382884, −6.16448714994801971560389408264, −5.37529250327508616478735763299, −3.55270267624104205177402583777, −2.91900417038703701652579305800, −1.44294068697220604739254303137, −0.56082995204069819894950510323,
1.50871916068412903754486393015, 2.76526458044318304276103213674, 3.64193846913668451350687605379, 4.98202740175258907458093604402, 5.55166394186694448282901757634, 7.10937463417480310586907971471, 7.56775005348758825516582414369, 8.665078775721175288161639746197, 9.324027780928940227781926629039, 9.818823404200563520479543649359
