| L(s) = 1 | + 38.2·2-s + 192.·3-s + 954.·4-s − 373.·5-s + 7.38e3·6-s + 1.11e4·7-s + 1.69e4·8-s + 1.74e4·9-s − 1.43e4·10-s + 1.75e4·11-s + 1.83e5·12-s + 4.30e4·13-s + 4.26e5·14-s − 7.20e4·15-s + 1.60e5·16-s + 3.89e5·17-s + 6.68e5·18-s − 5.55e5·19-s − 3.57e5·20-s + 2.14e6·21-s + 6.72e5·22-s − 1.12e6·23-s + 3.26e6·24-s − 1.81e6·25-s + 1.64e6·26-s − 4.30e5·27-s + 1.06e7·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.37·3-s + 1.86·4-s − 0.267·5-s + 2.32·6-s + 1.75·7-s + 1.46·8-s + 0.886·9-s − 0.452·10-s + 0.361·11-s + 2.56·12-s + 0.418·13-s + 2.96·14-s − 0.367·15-s + 0.612·16-s + 1.13·17-s + 1.50·18-s − 0.977·19-s − 0.498·20-s + 2.40·21-s + 0.611·22-s − 0.837·23-s + 2.01·24-s − 0.928·25-s + 0.707·26-s − 0.155·27-s + 3.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(12.46674529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.46674529\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 1.50e9T \) |
| good | 2 | \( 1 - 38.2T + 512T^{2} \) |
| 3 | \( 1 - 192.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 373.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.11e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.75e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.30e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.89e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.55e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.96e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.42e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.43e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.93e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.83e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.08e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.15e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.84e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.47e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.63e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.59e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.96e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.07e8T + 7.60e17T^{2} \) |
| 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
−11.37835452999820491203441290162, −9.955788357189764006058426763768, −8.198572512739574130440731490533, −8.118725191585102602113952146098, −6.57714217471735322169379669766, −5.27185765599963627473652209409, −4.27893195550284323676572350512, −3.56014085907631006428902859362, −2.36746045659270433535923868709, −1.52687843435521041926483355929,
1.52687843435521041926483355929, 2.36746045659270433535923868709, 3.56014085907631006428902859362, 4.27893195550284323676572350512, 5.27185765599963627473652209409, 6.57714217471735322169379669766, 8.118725191585102602113952146098, 8.198572512739574130440731490533, 9.955788357189764006058426763768, 11.37835452999820491203441290162
