| L(s) = 1 | + 2·3-s + 5-s + 7-s + 9-s + 4·13-s + 2·15-s − 4·19-s + 2·21-s − 6·23-s + 25-s − 4·27-s + 6·29-s + 4·31-s + 35-s + 2·37-s + 8·39-s + 6·41-s − 4·43-s + 45-s − 6·47-s + 49-s − 6·53-s − 8·57-s + 10·61-s + 63-s + 4·65-s − 2·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s − 0.917·19-s + 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s + 1.28·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.05·57-s + 1.28·61-s + 0.125·63-s + 0.496·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.28597238743858, −14.13665146007478, −13.43843267252598, −13.10514163641393, −12.64659582130093, −11.80144605379231, −11.48239949160913, −10.84733072338185, −10.20986219033638, −9.894621835898396, −9.251166048764284, −8.622944266543043, −8.311143124746890, −8.086478921231426, −7.236900096483750, −6.658587827366405, −5.895943386976498, −5.808976488312877, −4.655259865210874, −4.287560011960981, −3.660215631225761, −2.917515075399314, −2.514823736831793, −1.731702976356409, −1.247993792531031, 0,
1.247993792531031, 1.731702976356409, 2.514823736831793, 2.917515075399314, 3.660215631225761, 4.287560011960981, 4.655259865210874, 5.808976488312877, 5.895943386976498, 6.658587827366405, 7.236900096483750, 8.086478921231426, 8.311143124746890, 8.622944266543043, 9.251166048764284, 9.894621835898396, 10.20986219033638, 10.84733072338185, 11.48239949160913, 11.80144605379231, 12.64659582130093, 13.10514163641393, 13.43843267252598, 14.13665146007478, 14.28597238743858
