L(s) = 1 | − 2.30·3-s + 5-s + 7-s + 2.30·9-s + 2.60·11-s − 13-s − 2.30·15-s + 2.69·17-s − 2.69·19-s − 2.30·21-s − 8·23-s + 25-s + 1.60·27-s − 8.90·29-s − 4.30·31-s − 6·33-s + 35-s + 6.90·37-s + 2.30·39-s − 0.697·41-s − 2·43-s + 2.30·45-s + 2.60·47-s + 49-s − 6.21·51-s + 4.60·53-s + 2.60·55-s + ⋯ |
L(s) = 1 | − 1.32·3-s + 0.447·5-s + 0.377·7-s + 0.767·9-s + 0.785·11-s − 0.277·13-s − 0.594·15-s + 0.654·17-s − 0.618·19-s − 0.502·21-s − 1.66·23-s + 0.200·25-s + 0.308·27-s − 1.65·29-s − 0.772·31-s − 1.04·33-s + 0.169·35-s + 1.13·37-s + 0.368·39-s − 0.108·41-s − 0.304·43-s + 0.343·45-s + 0.380·47-s + 0.142·49-s − 0.869·51-s + 0.632·53-s + 0.351·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 8.90T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 - 6.90T + 37T^{2} \) |
| 41 | \( 1 + 0.697T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 4.60T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 1.90T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 + 1.39T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 9.81T + 83T^{2} \) |
| 89 | \( 1 + 3.69T + 89T^{2} \) |
| 97 | \( 1 - 1.21T + 97T^{2} \) |
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
−8.067464210458339668810120020108, −7.31479233866719182303397020267, −6.46321376564340113773256675754, −5.85538857997527870608778738646, −5.39380117047436735248037978195, −4.43453738527684569921546182870, −3.71462676586366903041159702243, −2.25525725598273649510170015073, −1.32561580890176224184356607069, 0,
1.32561580890176224184356607069, 2.25525725598273649510170015073, 3.71462676586366903041159702243, 4.43453738527684569921546182870, 5.39380117047436735248037978195, 5.85538857997527870608778738646, 6.46321376564340113773256675754, 7.31479233866719182303397020267, 8.067464210458339668810120020108