| L(s) = 1 | + 140.·3-s − 1.23e3·5-s + 2.40e3·7-s + 162.·9-s − 1.19e4·11-s + 7.05e4·13-s − 1.73e5·15-s + 3.52e5·17-s − 3.71e5·19-s + 3.38e5·21-s − 3.42e5·23-s − 4.28e5·25-s − 2.74e6·27-s + 1.67e6·29-s − 6.38e6·31-s − 1.67e6·33-s − 2.96e6·35-s − 2.11e7·37-s + 9.94e6·39-s − 1.97e7·41-s + 1.65e7·43-s − 2.00e5·45-s − 1.70e7·47-s + 5.76e6·49-s + 4.97e7·51-s + 6.53e7·53-s + 1.47e7·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 0.883·5-s + 0.377·7-s + 0.00823·9-s − 0.245·11-s + 0.685·13-s − 0.887·15-s + 1.02·17-s − 0.654·19-s + 0.379·21-s − 0.255·23-s − 0.219·25-s − 0.995·27-s + 0.439·29-s − 1.24·31-s − 0.246·33-s − 0.333·35-s − 1.85·37-s + 0.688·39-s − 1.09·41-s + 0.738·43-s − 0.00727·45-s − 0.511·47-s + 0.142·49-s + 1.02·51-s + 1.13·53-s + 0.216·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 3 | \( 1 - 140.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.23e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 1.19e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.05e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.52e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.71e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.42e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.11e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.97e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.65e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.70e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.53e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.92e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.48e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.24e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.09e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.24e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.00e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.96e8T + 7.60e17T^{2} \) |
| 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
−11.37211960482704135705723988769, −10.24754384706212476328066882176, −8.820419105710141973854788349581, −8.195351295409218943987048474653, −7.26292710836075734876161836585, −5.59117322943696552248326465461, −4.02867133881979766326343764035, −3.15244372213429581802583135762, −1.70219529909319786190418957379, 0,
1.70219529909319786190418957379, 3.15244372213429581802583135762, 4.02867133881979766326343764035, 5.59117322943696552248326465461, 7.26292710836075734876161836585, 8.195351295409218943987048474653, 8.820419105710141973854788349581, 10.24754384706212476328066882176, 11.37211960482704135705723988769
