L(s) = 1 | + 2-s − 3-s + 4-s + 1.40·5-s − 6-s − 3.03·7-s + 8-s + 9-s + 1.40·10-s − 11-s − 12-s + 3.73·13-s − 3.03·14-s − 1.40·15-s + 16-s − 7.21·17-s + 18-s + 6.74·19-s + 1.40·20-s + 3.03·21-s − 22-s − 5.31·23-s − 24-s − 3.01·25-s + 3.73·26-s − 27-s − 3.03·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.629·5-s − 0.408·6-s − 1.14·7-s + 0.353·8-s + 0.333·9-s + 0.445·10-s − 0.301·11-s − 0.288·12-s + 1.03·13-s − 0.810·14-s − 0.363·15-s + 0.250·16-s − 1.74·17-s + 0.235·18-s + 1.54·19-s + 0.314·20-s + 0.662·21-s − 0.213·22-s − 1.10·23-s − 0.204·24-s − 0.603·25-s + 0.732·26-s − 0.192·27-s − 0.573·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 5.31T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 0.0646T + 31T^{2} \) |
| 37 | \( 1 - 1.89T + 37T^{2} \) |
| 41 | \( 1 + 7.30T + 41T^{2} \) |
| 43 | \( 1 - 9.52T + 43T^{2} \) |
| 47 | \( 1 - 4.75T + 47T^{2} \) |
| 53 | \( 1 + 7.05T + 53T^{2} \) |
| 59 | \( 1 - 2.27T + 59T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 - 6.68T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 2.10T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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
−7.85240165987814252996831967017, −7.09121341464750194927906182823, −6.32390770847037434344760124949, −5.89517625600045649030421348086, −5.30411238458266671259298008227, −4.17700260136035382122556047690, −3.58372307040851870079948599879, −2.54865183163965335659207529303, −1.58346327313316676378374994541, 0,
1.58346327313316676378374994541, 2.54865183163965335659207529303, 3.58372307040851870079948599879, 4.17700260136035382122556047690, 5.30411238458266671259298008227, 5.89517625600045649030421348086, 6.32390770847037434344760124949, 7.09121341464750194927906182823, 7.85240165987814252996831967017