L(s) = 1 | + (−3.23 − 3.23i)7-s − 4.57i·11-s + (4.23 − 4.23i)13-s + (1.74 − 1.74i)17-s + 2.47i·19-s + (−2.82 − 2.82i)23-s − 5.99·29-s − 1.52·31-s + (−2.23 − 2.23i)37-s + 7.07i·41-s + (2.47 − 2.47i)43-s + (1.74 − 1.74i)47-s + 13.9i·49-s + 1.08·59-s + 10.4·61-s + ⋯ |
L(s) = 1 | + (−1.22 − 1.22i)7-s − 1.37i·11-s + (1.17 − 1.17i)13-s + (0.423 − 0.423i)17-s + 0.567i·19-s + (−0.589 − 0.589i)23-s − 1.11·29-s − 0.274·31-s + (−0.367 − 0.367i)37-s + 1.10i·41-s + (0.376 − 0.376i)43-s + (0.254 − 0.254i)47-s + 1.99i·49-s + 0.140·59-s + 1.34·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9378753669\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9378753669\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3.23 + 3.23i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.57iT - 11T^{2} \) |
| 13 | \( 1 + (-4.23 + 4.23i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.74 + 1.74i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.47iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + (2.23 + 2.23i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-2.47 + 2.47i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.74 + 1.74i)T - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + (1.52 + 1.52i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (-0.527 + 0.527i)T - 73iT^{2} \) |
| 79 | \( 1 + 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (1.08 + 1.08i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.746T + 89T^{2} \) |
| 97 | \( 1 + (1 + i)T + 97iT^{2} \) |
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
−8.134742455777848600732805387095, −7.51894284228158324804447073234, −6.60422831080188628368382789234, −5.98471142495353414342854051223, −5.40570677644849095877746029554, −3.90598080644699329473097623558, −3.61549950742967848936014059394, −2.82616829718916339900585235393, −1.12859100393214578229272226660, −0.30363558197725891493714776578,
1.66272567367332962005250794462, 2.40668451642008303339672532144, 3.53111925359335885314747336777, 4.15654268613470361667224460598, 5.31823585457151582445623441982, 5.95484662604304644272696939506, 6.67839954944379568572636549513, 7.26060247340918086948321735461, 8.320662247462532104139613028212, 9.097963178579602245722023735945