L(s) = 1 | + 2-s + 3-s + 4-s − 2.21·5-s + 6-s + 8-s + 9-s − 2.21·10-s + 3.95·11-s + 12-s − 7.06·13-s − 2.21·15-s + 16-s − 5.21·17-s + 18-s − 19-s − 2.21·20-s + 3.95·22-s + 7.23·23-s + 24-s − 0.112·25-s − 7.06·26-s + 27-s + 1.11·29-s − 2.21·30-s − 2.57·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.988·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.699·10-s + 1.19·11-s + 0.288·12-s − 1.96·13-s − 0.570·15-s + 0.250·16-s − 1.26·17-s + 0.235·18-s − 0.229·19-s − 0.494·20-s + 0.843·22-s + 1.50·23-s + 0.204·24-s − 0.0225·25-s − 1.38·26-s + 0.192·27-s + 0.206·29-s − 0.403·30-s − 0.463·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2.21T + 5T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + 7.06T + 13T^{2} \) |
| 17 | \( 1 + 5.21T + 17T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 - 1.11T + 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 + 5.85T + 37T^{2} \) |
| 41 | \( 1 + 2.67T + 41T^{2} \) |
| 43 | \( 1 + 9.76T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 8.06T + 53T^{2} \) |
| 59 | \( 1 + 4.32T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 - 1.06T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 8.02T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 2.67T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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
−7.55242429197481531324431559922, −7.03193611717033617997961523063, −6.67035919036105583355916893509, −5.40747554217990960495386576465, −4.59184990633307491702847922812, −4.20357014294609022577016505427, −3.31456174699558354174212799672, −2.58799745055963922782901793139, −1.60983465963788798802400960030, 0,
1.60983465963788798802400960030, 2.58799745055963922782901793139, 3.31456174699558354174212799672, 4.20357014294609022577016505427, 4.59184990633307491702847922812, 5.40747554217990960495386576465, 6.67035919036105583355916893509, 7.03193611717033617997961523063, 7.55242429197481531324431559922