| L(s) = 1 | + 43.8·2-s + 1.41e3·4-s + 7.86e3·7-s + 3.95e4·8-s + 4.93e4·11-s − 2.42e4·13-s + 3.44e5·14-s + 1.01e6·16-s + 2.68e5·17-s − 1.68e5·19-s + 2.16e6·22-s − 2.12e6·23-s − 1.06e6·26-s + 1.11e7·28-s − 3.89e5·29-s + 9.05e4·31-s + 2.41e7·32-s + 1.17e7·34-s + 3.31e6·37-s − 7.38e6·38-s − 2.32e7·41-s − 1.91e7·43-s + 6.98e7·44-s − 9.31e7·46-s + 6.28e7·47-s + 2.14e7·49-s − 3.42e7·52-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.76·4-s + 1.23·7-s + 3.41·8-s + 1.01·11-s − 0.235·13-s + 2.40·14-s + 3.86·16-s + 0.778·17-s − 0.296·19-s + 1.97·22-s − 1.58·23-s − 0.456·26-s + 3.41·28-s − 0.102·29-s + 0.0176·31-s + 4.07·32-s + 1.51·34-s + 0.291·37-s − 0.574·38-s − 1.28·41-s − 0.852·43-s + 2.80·44-s − 3.06·46-s + 1.87·47-s + 0.531·49-s − 0.650·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(11.13680932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.13680932\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 43.8T + 512T^{2} \) |
| 7 | \( 1 - 7.86e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.93e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.42e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.68e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.89e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.05e4T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.31e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.91e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.28e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.80e5T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.84e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.53e5T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.39e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.28e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.39e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.28e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.12e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.07e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.70e9T + 7.60e17T^{2} \) |
| 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
−11.13273510292471023562818767264, −10.07054311986075153218788592540, −8.261306065617549893854654271662, −7.29954942395982781936269700194, −6.23235683213134233586545329347, −5.32461757183912240518117820847, −4.39148124738874526125396584609, −3.60599763762996168133480229799, −2.21104699988800856367735432181, −1.37092736223913751605471797823,
1.37092736223913751605471797823, 2.21104699988800856367735432181, 3.60599763762996168133480229799, 4.39148124738874526125396584609, 5.32461757183912240518117820847, 6.23235683213134233586545329347, 7.29954942395982781936269700194, 8.261306065617549893854654271662, 10.07054311986075153218788592540, 11.13273510292471023562818767264
