| L(s) = 1 | + 1.17e4·5-s + 1.68e4·7-s + 5.59e5·11-s + 4.79e5·13-s + 1.05e7·17-s − 1.66e7·19-s + 1.69e7·23-s + 8.94e7·25-s + 1.71e7·29-s + 1.46e8·31-s + 1.97e8·35-s − 7.01e8·37-s + 6.75e8·41-s + 4.83e8·43-s + 1.93e9·47-s + 2.82e8·49-s + 1.39e9·53-s + 6.58e9·55-s + 6.21e9·59-s − 1.08e10·61-s + 5.63e9·65-s + 6.71e9·67-s + 1.98e10·71-s − 1.83e10·73-s + 9.40e9·77-s − 4.21e10·79-s + 1.13e10·83-s + ⋯ |
| L(s) = 1 | + 1.68·5-s + 0.377·7-s + 1.04·11-s + 0.357·13-s + 1.80·17-s − 1.54·19-s + 0.550·23-s + 1.83·25-s + 0.154·29-s + 0.918·31-s + 0.636·35-s − 1.66·37-s + 0.910·41-s + 0.501·43-s + 1.22·47-s + 0.142·49-s + 0.456·53-s + 1.76·55-s + 1.13·59-s − 1.64·61-s + 0.602·65-s + 0.607·67-s + 1.30·71-s − 1.03·73-s + 0.396·77-s − 1.54·79-s + 0.315·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.536558203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.536558203\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 5 | \( 1 - 1.17e4T + 4.88e7T^{2} \) |
| 11 | \( 1 - 5.59e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 4.79e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.05e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.66e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.69e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.71e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.46e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.01e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 6.75e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.83e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.93e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.39e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 6.21e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 1.08e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 6.71e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.98e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.83e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.21e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.13e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.17e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.35e11T + 7.15e21T^{2} \) |
| 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
−10.09855953664800320810911548733, −9.209848134791955895043841362259, −8.415917991747834027155648774162, −6.97594378346142507681394519184, −6.07776591273526949000467990884, −5.37563984055274464746499785108, −4.09756558566626960370871361530, −2.74905799682261324192839494911, −1.66999210249605074879239296480, −0.994469818751355916729101620495,
0.994469818751355916729101620495, 1.66999210249605074879239296480, 2.74905799682261324192839494911, 4.09756558566626960370871361530, 5.37563984055274464746499785108, 6.07776591273526949000467990884, 6.97594378346142507681394519184, 8.415917991747834027155648774162, 9.209848134791955895043841362259, 10.09855953664800320810911548733
