L(s) = 1 | − 2.56·2-s − 3-s + 4.56·4-s + 5-s + 2.56·6-s − 7-s − 6.56·8-s + 9-s − 2.56·10-s + 11-s − 4.56·12-s + 1.12·13-s + 2.56·14-s − 15-s + 7.68·16-s − 3.56·17-s − 2.56·18-s + 1.56·19-s + 4.56·20-s + 21-s − 2.56·22-s − 5.56·23-s + 6.56·24-s + 25-s − 2.87·26-s − 27-s − 4.56·28-s + ⋯ |
L(s) = 1 | − 1.81·2-s − 0.577·3-s + 2.28·4-s + 0.447·5-s + 1.04·6-s − 0.377·7-s − 2.31·8-s + 0.333·9-s − 0.810·10-s + 0.301·11-s − 1.31·12-s + 0.311·13-s + 0.684·14-s − 0.258·15-s + 1.92·16-s − 0.863·17-s − 0.603·18-s + 0.358·19-s + 1.01·20-s + 0.218·21-s − 0.546·22-s − 1.15·23-s + 1.33·24-s + 0.200·25-s − 0.564·26-s − 0.192·27-s − 0.862·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 13 | \( 1 - 1.12T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 1.56T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 1.31T + 53T^{2} \) |
| 59 | \( 1 - 6.43T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 + 2.87T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 + 9.56T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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
−9.513967122264208968458849986444, −8.739180470391774304001035876449, −7.86183836477138691924054528738, −7.00023991855283618727054549133, −6.33124672664491325866903660357, −5.55684075551602445238977563965, −3.96731816426904121676312942968, −2.45959341744008594917124935014, −1.42753918033326313510965428364, 0,
1.42753918033326313510965428364, 2.45959341744008594917124935014, 3.96731816426904121676312942968, 5.55684075551602445238977563965, 6.33124672664491325866903660357, 7.00023991855283618727054549133, 7.86183836477138691924054528738, 8.739180470391774304001035876449, 9.513967122264208968458849986444