| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 2·11-s − 12-s − 4·13-s + 16-s + 2·17-s + 18-s + 2·22-s + 6·23-s − 24-s − 4·26-s − 27-s − 2·29-s − 4·31-s + 32-s − 2·33-s + 2·34-s + 36-s + 8·37-s + 4·39-s + 6·41-s + 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.426·22-s + 1.25·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/6·36-s + 1.31·37-s + 0.640·39-s + 0.937·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.459384219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.459384219\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| 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
−14.52953581533215, −13.99176799933089, −13.31238376532599, −12.76143753690445, −12.49639631928544, −11.95471411612703, −11.40664037866582, −10.96933523166047, −10.54018791153904, −9.710468487194412, −9.422250886355486, −8.837122884534181, −7.828626634207944, −7.549169615229192, −6.921728394388070, −6.456332334123669, −5.758812821128451, −5.326674537540918, −4.778324085861320, −4.149214817856526, −3.649024313854359, −2.762896347796317, −2.291550877802241, −1.310437083732597, −0.6326202869765111,
0.6326202869765111, 1.310437083732597, 2.291550877802241, 2.762896347796317, 3.649024313854359, 4.149214817856526, 4.778324085861320, 5.326674537540918, 5.758812821128451, 6.456332334123669, 6.921728394388070, 7.549169615229192, 7.828626634207944, 8.837122884534181, 9.422250886355486, 9.710468487194412, 10.54018791153904, 10.96933523166047, 11.40664037866582, 11.95471411612703, 12.49639631928544, 12.76143753690445, 13.31238376532599, 13.99176799933089, 14.52953581533215
