| L(s) = 1 | + 243·3-s + 5.76e3·5-s − 7.24e4·7-s + 5.90e4·9-s + 4.08e5·11-s + 1.36e6·13-s + 1.40e6·15-s + 5.42e6·17-s − 1.51e7·19-s − 1.76e7·21-s + 5.21e7·23-s − 1.55e7·25-s + 1.43e7·27-s + 1.18e8·29-s + 5.76e7·31-s + 9.93e7·33-s − 4.17e8·35-s − 3.75e8·37-s + 3.32e8·39-s + 8.56e8·41-s + 1.24e9·43-s + 3.40e8·45-s + 1.30e9·47-s + 3.27e9·49-s + 1.31e9·51-s + 4.09e8·53-s + 2.35e9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.825·5-s − 1.62·7-s + 1/3·9-s + 0.765·11-s + 1.02·13-s + 0.476·15-s + 0.926·17-s − 1.40·19-s − 0.940·21-s + 1.69·23-s − 0.319·25-s + 0.192·27-s + 1.07·29-s + 0.361·31-s + 0.442·33-s − 1.34·35-s − 0.891·37-s + 0.589·39-s + 1.15·41-s + 1.29·43-s + 0.275·45-s + 0.831·47-s + 1.65·49-s + 0.534·51-s + 0.134·53-s + 0.631·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.745350173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.745350173\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 - 5766 T + p^{11} T^{2} \) |
| 7 | \( 1 + 10352 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 408948 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1367558 T + p^{11} T^{2} \) |
| 17 | \( 1 - 5422914 T + p^{11} T^{2} \) |
| 19 | \( 1 + 15166100 T + p^{11} T^{2} \) |
| 23 | \( 1 - 52194072 T + p^{11} T^{2} \) |
| 29 | \( 1 - 118581150 T + p^{11} T^{2} \) |
| 31 | \( 1 - 57652408 T + p^{11} T^{2} \) |
| 37 | \( 1 + 375985186 T + p^{11} T^{2} \) |
| 41 | \( 1 - 856316202 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1245189172 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1306762656 T + p^{11} T^{2} \) |
| 53 | \( 1 - 409556358 T + p^{11} T^{2} \) |
| 59 | \( 1 - 48862140 p T + p^{11} T^{2} \) |
| 61 | \( 1 - 5731767302 T + p^{11} T^{2} \) |
| 67 | \( 1 + 3893272244 T + p^{11} T^{2} \) |
| 71 | \( 1 - 9075890088 T + p^{11} T^{2} \) |
| 73 | \( 1 + 15571822822 T + p^{11} T^{2} \) |
| 79 | \( 1 - 30196762600 T + p^{11} T^{2} \) |
| 83 | \( 1 + 23135252628 T + p^{11} T^{2} \) |
| 89 | \( 1 + 25614819990 T + p^{11} T^{2} \) |
| 97 | \( 1 + 61937553406 T + p^{11} T^{2} \) |
| 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
−13.28164073879688585818033849288, −12.46998202997260884015824817911, −10.60822579251143985262507462778, −9.554787467981749460580054592166, −8.724808877926491824026990853031, −6.83404527089779301142648615705, −5.95089175019307506437847175296, −3.85788867168345826247058833875, −2.68008753493332759712008446861, −1.02105613256072245672238324416,
1.02105613256072245672238324416, 2.68008753493332759712008446861, 3.85788867168345826247058833875, 5.95089175019307506437847175296, 6.83404527089779301142648615705, 8.724808877926491824026990853031, 9.554787467981749460580054592166, 10.60822579251143985262507462778, 12.46998202997260884015824817911, 13.28164073879688585818033849288
