L(s) = 1 | + 2-s + 3-s + 4-s − 3.80·5-s + 6-s − 0.753·7-s + 8-s + 9-s − 3.80·10-s + 3.24·11-s + 12-s − 4.29·13-s − 0.753·14-s − 3.80·15-s + 16-s + 2.85·17-s + 18-s + 19-s − 3.80·20-s − 0.753·21-s + 3.24·22-s − 0.225·23-s + 24-s + 9.45·25-s − 4.29·26-s + 27-s − 0.753·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.70·5-s + 0.408·6-s − 0.284·7-s + 0.353·8-s + 0.333·9-s − 1.20·10-s + 0.979·11-s + 0.288·12-s − 1.19·13-s − 0.201·14-s − 0.981·15-s + 0.250·16-s + 0.691·17-s + 0.235·18-s + 0.229·19-s − 0.850·20-s − 0.164·21-s + 0.692·22-s − 0.0469·23-s + 0.204·24-s + 1.89·25-s − 0.842·26-s + 0.192·27-s − 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 + 0.753T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 23 | \( 1 + 0.225T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 + 6.27T + 31T^{2} \) |
| 37 | \( 1 - 0.631T + 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 - 6.04T + 43T^{2} \) |
| 47 | \( 1 + 6.82T + 47T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 + 4.30T + 61T^{2} \) |
| 67 | \( 1 - 6.61T + 67T^{2} \) |
| 71 | \( 1 + 0.121T + 71T^{2} \) |
| 73 | \( 1 + 4.67T + 73T^{2} \) |
| 79 | \( 1 - 4.00T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 6.84T + 97T^{2} \) |
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
−7.69510328556869651231963503531, −7.11093657970390347709241314004, −6.49004255498610055479225739186, −5.35360662843117230550185456922, −4.63748381785000333273376840690, −3.85827572131819567479539257335, −3.48919580073189489310266632415, −2.67393406657497866352544998172, −1.44340106237141818202854656246, 0,
1.44340106237141818202854656246, 2.67393406657497866352544998172, 3.48919580073189489310266632415, 3.85827572131819567479539257335, 4.63748381785000333273376840690, 5.35360662843117230550185456922, 6.49004255498610055479225739186, 7.11093657970390347709241314004, 7.69510328556869651231963503531