L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s + 13-s + 6·19-s + 2·21-s + 6·23-s − 27-s + 8·29-s + 4·33-s + 10·37-s − 39-s − 10·41-s − 12·43-s + 12·47-s − 3·49-s − 6·53-s − 6·57-s − 12·59-s + 10·61-s − 2·63-s − 4·67-s − 6·69-s − 8·71-s + 4·73-s + 8·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.37·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s + 1.48·29-s + 0.696·33-s + 1.64·37-s − 0.160·39-s − 1.56·41-s − 1.82·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.794·57-s − 1.56·59-s + 1.28·61-s − 0.251·63-s − 0.488·67-s − 0.722·69-s − 0.949·71-s + 0.468·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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
−14.54298818236138, −13.80426733479689, −13.43096566564246, −13.05081592935754, −12.55631788342714, −11.91898871930918, −11.58353181917817, −10.91199404439652, −10.46435597793716, −9.964491095014484, −9.573166905578155, −8.912970363165595, −8.273196548471667, −7.757885876220729, −7.146433558918232, −6.670939885146735, −6.117961026831186, −5.503321288491777, −4.976880519273051, −4.586106055543461, −3.614201293596565, −2.985445377874757, −2.695896579114221, −1.523910977904752, −0.8475481143952718, 0,
0.8475481143952718, 1.523910977904752, 2.695896579114221, 2.985445377874757, 3.614201293596565, 4.586106055543461, 4.976880519273051, 5.503321288491777, 6.117961026831186, 6.670939885146735, 7.146433558918232, 7.757885876220729, 8.273196548471667, 8.912970363165595, 9.573166905578155, 9.964491095014484, 10.46435597793716, 10.91199404439652, 11.58353181917817, 11.91898871930918, 12.55631788342714, 13.05081592935754, 13.43096566564246, 13.80426733479689, 14.54298818236138