L(s) = 1 | − 0.239·2-s − 1.94·4-s − 1.18·5-s + 0.942·8-s + 0.282·10-s − 3.70·11-s − 13-s + 3.66·16-s + 6.94·17-s − 1.94·19-s + 2.29·20-s + 0.885·22-s − 5.60·23-s − 3.60·25-s + 0.239·26-s + 0.239·29-s − 1.66·31-s − 2.76·32-s − 1.66·34-s − 9.54·37-s + 0.464·38-s − 1.11·40-s + 10.1·41-s + 2.22·43-s + 7.19·44-s + 1.33·46-s − 5.82·47-s + ⋯ |
L(s) = 1 | − 0.169·2-s − 0.971·4-s − 0.528·5-s + 0.333·8-s + 0.0893·10-s − 1.11·11-s − 0.277·13-s + 0.915·16-s + 1.68·17-s − 0.445·19-s + 0.513·20-s + 0.188·22-s − 1.16·23-s − 0.720·25-s + 0.0468·26-s + 0.0444·29-s − 0.298·31-s − 0.488·32-s − 0.284·34-s − 1.56·37-s + 0.0753·38-s − 0.176·40-s + 1.59·41-s + 0.339·43-s + 1.08·44-s + 0.197·46-s − 0.850·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6992302176\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6992302176\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.239T + 2T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 6.94T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 + 5.60T + 23T^{2} \) |
| 29 | \( 1 - 0.239T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 - 7.60T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 - 8.60T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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
−8.224109815599994742633414242189, −7.907399927394722797008715240728, −7.34412457970403397876604625464, −6.04208313840883713592509414739, −5.41836041405943411281137064347, −4.69968305679906952629261741620, −3.84391940630156707376799032610, −3.16543783066093183300898347644, −1.88119015681072879994872662474, −0.48722328896480827272175970429,
0.48722328896480827272175970429, 1.88119015681072879994872662474, 3.16543783066093183300898347644, 3.84391940630156707376799032610, 4.69968305679906952629261741620, 5.41836041405943411281137064347, 6.04208313840883713592509414739, 7.34412457970403397876604625464, 7.907399927394722797008715240728, 8.224109815599994742633414242189