L(s) = 1 | − 1.01·2-s − 1.33·3-s − 0.975·4-s + 3.44·5-s + 1.34·6-s + 1.50·7-s + 3.01·8-s − 1.22·9-s − 3.49·10-s + 11-s + 1.29·12-s + 5.15·13-s − 1.52·14-s − 4.59·15-s − 1.09·16-s + 3.32·17-s + 1.24·18-s − 1.84·19-s − 3.36·20-s − 2.00·21-s − 1.01·22-s + 5.95·23-s − 4.01·24-s + 6.89·25-s − 5.21·26-s + 5.62·27-s − 1.46·28-s + ⋯ |
L(s) = 1 | − 0.715·2-s − 0.769·3-s − 0.487·4-s + 1.54·5-s + 0.550·6-s + 0.568·7-s + 1.06·8-s − 0.408·9-s − 1.10·10-s + 0.301·11-s + 0.375·12-s + 1.42·13-s − 0.406·14-s − 1.18·15-s − 0.274·16-s + 0.807·17-s + 0.292·18-s − 0.423·19-s − 0.752·20-s − 0.436·21-s − 0.215·22-s + 1.24·23-s − 0.819·24-s + 1.37·25-s − 1.02·26-s + 1.08·27-s − 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.165952488\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.165952488\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 2 | \( 1 + 1.01T + 2T^{2} \) |
| 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 + 1.84T + 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 + 4.75T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 0.187T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 9.24T + 59T^{2} \) |
| 61 | \( 1 + 7.30T + 61T^{2} \) |
| 67 | \( 1 - 7.00T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 3.06T + 73T^{2} \) |
| 79 | \( 1 - 4.22T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 - 2.37T + 97T^{2} \) |
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
−9.415549152850947535103089228465, −8.861473019096130702326029033598, −8.245540056750322173425202861072, −7.02583506310523933105319433149, −6.10417458860214313992859027749, −5.48153563403949546377909908904, −4.81682389895439939964907394666, −3.44254656754891460534668919223, −1.83217526673657451180109686148, −0.986048953168040130617673845101,
0.986048953168040130617673845101, 1.83217526673657451180109686148, 3.44254656754891460534668919223, 4.81682389895439939964907394666, 5.48153563403949546377909908904, 6.10417458860214313992859027749, 7.02583506310523933105319433149, 8.245540056750322173425202861072, 8.861473019096130702326029033598, 9.415549152850947535103089228465