L(s) = 1 | − 2-s − 3-s + 4-s + 2.19·5-s + 6-s + 1.89·7-s − 8-s + 9-s − 2.19·10-s + 11-s − 12-s + 0.371·13-s − 1.89·14-s − 2.19·15-s + 16-s + 6.08·17-s − 18-s + 0.286·19-s + 2.19·20-s − 1.89·21-s − 22-s − 2.45·23-s + 24-s − 0.186·25-s − 0.371·26-s − 27-s + 1.89·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.981·5-s + 0.408·6-s + 0.717·7-s − 0.353·8-s + 0.333·9-s − 0.693·10-s + 0.301·11-s − 0.288·12-s + 0.102·13-s − 0.507·14-s − 0.566·15-s + 0.250·16-s + 1.47·17-s − 0.235·18-s + 0.0657·19-s + 0.490·20-s − 0.414·21-s − 0.213·22-s − 0.512·23-s + 0.204·24-s − 0.0372·25-s − 0.0728·26-s − 0.192·27-s + 0.358·28-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
\(L(1)\) |
\(\approx\) |
\(1.686299424\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.686299424\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.19T + 5T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 13 | \( 1 - 0.371T + 13T^{2} \) |
| 17 | \( 1 - 6.08T + 17T^{2} \) |
| 19 | \( 1 - 0.286T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 - 4.95T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 - 9.98T + 43T^{2} \) |
| 47 | \( 1 - 3.88T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 + 1.14T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.385946803708197768026501192832, −7.81678658687732603383553412779, −6.99913546980430298731917607895, −6.15985907858953918122083823568, −5.66348162769541764286597838880, −4.91096294881937951856852107033, −3.85077192205381573842095945530, −2.65762202794056096400243566574, −1.68167050462008812849459547992, −0.922462178476051863409495181789,
0.922462178476051863409495181789, 1.68167050462008812849459547992, 2.65762202794056096400243566574, 3.85077192205381573842095945530, 4.91096294881937951856852107033, 5.66348162769541764286597838880, 6.15985907858953918122083823568, 6.99913546980430298731917607895, 7.81678658687732603383553412779, 8.385946803708197768026501192832