| L(s) = 1 | + 2-s − 2.22·3-s + 4-s + 1.44·5-s − 2.22·6-s + 1.32·7-s + 8-s + 1.95·9-s + 1.44·10-s − 4.69·11-s − 2.22·12-s + 1.27·13-s + 1.32·14-s − 3.21·15-s + 16-s − 1.75·17-s + 1.95·18-s − 1.83·19-s + 1.44·20-s − 2.93·21-s − 4.69·22-s + 1.61·23-s − 2.22·24-s − 2.90·25-s + 1.27·26-s + 2.32·27-s + 1.32·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.28·3-s + 0.5·4-s + 0.646·5-s − 0.908·6-s + 0.499·7-s + 0.353·8-s + 0.651·9-s + 0.457·10-s − 1.41·11-s − 0.642·12-s + 0.354·13-s + 0.352·14-s − 0.831·15-s + 0.250·16-s − 0.426·17-s + 0.460·18-s − 0.421·19-s + 0.323·20-s − 0.641·21-s − 1.00·22-s + 0.336·23-s − 0.454·24-s − 0.581·25-s + 0.251·26-s + 0.447·27-s + 0.249·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 4021 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.22T + 3T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 17 | \( 1 + 1.75T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 + 1.67T + 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 + 6.43T + 53T^{2} \) |
| 59 | \( 1 - 1.32T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 0.908T + 73T^{2} \) |
| 79 | \( 1 + 4.37T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 9.16T + 97T^{2} \) |
| 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
−7.23494791199863499297811054643, −6.46419368647411581232041902464, −6.02484398908067198412316151233, −5.30667403965495693910056544173, −4.93161016900530519400779801004, −4.24684238180660547700264362434, −3.03849519042708287908227121478, −2.28278652826600477655238612042, −1.29422821541622375755855330043, 0,
1.29422821541622375755855330043, 2.28278652826600477655238612042, 3.03849519042708287908227121478, 4.24684238180660547700264362434, 4.93161016900530519400779801004, 5.30667403965495693910056544173, 6.02484398908067198412316151233, 6.46419368647411581232041902464, 7.23494791199863499297811054643
