| L(s) = 1 | − 1.79·2-s + 0.867·3-s − 0.768·4-s − 1.56·6-s − 7·7-s + 8.57·8-s − 8.24·9-s − 0.666·12-s + 25.1·13-s + 12.5·14-s − 12.3·16-s + 6.49·17-s + 14.8·18-s − 29.1·19-s − 6.07·21-s + 45.9·23-s + 7.43·24-s + 25·25-s − 45.2·26-s − 14.9·27-s + 5.37·28-s − 12.1·32-s − 11.6·34-s + 6.33·36-s + 53.9·37-s + 52.3·38-s + 21.8·39-s + ⋯ |
| L(s) = 1 | − 0.898·2-s + 0.289·3-s − 0.192·4-s − 0.260·6-s − 7-s + 1.07·8-s − 0.916·9-s − 0.0555·12-s + 1.93·13-s + 0.898·14-s − 0.771·16-s + 0.382·17-s + 0.823·18-s − 1.53·19-s − 0.289·21-s + 1.99·23-s + 0.309·24-s + 25-s − 1.74·26-s − 0.554·27-s + 0.192·28-s − 0.378·32-s − 0.343·34-s + 0.175·36-s + 1.45·37-s + 1.37·38-s + 0.560·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8530325019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8530325019\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 + 1.79T + 4T^{2} \) |
| 3 | \( 1 - 0.867T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 25.1T + 169T^{2} \) |
| 17 | \( 1 - 6.49T + 289T^{2} \) |
| 19 | \( 1 + 29.1T + 361T^{2} \) |
| 23 | \( 1 - 45.9T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 53.9T + 1.36e3T^{2} \) |
| 43 | \( 1 - 61.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 17.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 161.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 71.9T + 9.40e3T^{2} \) |
| 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
−11.05500902617162709538466035275, −10.73627124860696147766389226191, −9.347258543321586476417366180436, −8.863618292103984462910378309926, −8.132771989303281965098542349144, −6.78317319066224765343750699449, −5.78848884912009002408233154321, −4.16059233239379040205061914312, −2.90935211523299742141256627500, −0.879332753478647521789236891062,
0.879332753478647521789236891062, 2.90935211523299742141256627500, 4.16059233239379040205061914312, 5.78848884912009002408233154321, 6.78317319066224765343750699449, 8.132771989303281965098542349144, 8.863618292103984462910378309926, 9.347258543321586476417366180436, 10.73627124860696147766389226191, 11.05500902617162709538466035275
