L(s) = 1 | + 2.45·3-s + 2.74·5-s + 0.283·7-s + 3.00·9-s + 4.12·11-s + 0.0271·13-s + 6.71·15-s − 1.28·17-s + 5.72·19-s + 0.695·21-s + 5.97·23-s + 2.51·25-s + 0.0132·27-s − 2.55·29-s + 2.02·31-s + 10.1·33-s + 0.777·35-s + 2.42·37-s + 0.0666·39-s − 11.0·41-s − 10.4·43-s + 8.24·45-s − 4.54·47-s − 6.91·49-s − 3.15·51-s − 9.30·53-s + 11.3·55-s + ⋯ |
L(s) = 1 | + 1.41·3-s + 1.22·5-s + 0.107·7-s + 1.00·9-s + 1.24·11-s + 0.00754·13-s + 1.73·15-s − 0.312·17-s + 1.31·19-s + 0.151·21-s + 1.24·23-s + 0.503·25-s + 0.00254·27-s − 0.474·29-s + 0.364·31-s + 1.76·33-s + 0.131·35-s + 0.399·37-s + 0.0106·39-s − 1.72·41-s − 1.60·43-s + 1.22·45-s − 0.662·47-s − 0.988·49-s − 0.441·51-s − 1.27·53-s + 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.573377107\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.573377107\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 241 | \( 1 + T \) |
good | 3 | \( 1 - 2.45T + 3T^{2} \) |
| 5 | \( 1 - 2.74T + 5T^{2} \) |
| 7 | \( 1 - 0.283T + 7T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 - 0.0271T + 13T^{2} \) |
| 17 | \( 1 + 1.28T + 17T^{2} \) |
| 19 | \( 1 - 5.72T + 19T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 - 2.02T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 4.54T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 - 9.94T + 59T^{2} \) |
| 61 | \( 1 - 8.17T + 61T^{2} \) |
| 67 | \( 1 + 4.40T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 6.69T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 - 0.746T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.704002151935151580773315836280, −7.893113182094629940308779370867, −6.98700130116792239690493227485, −6.45910956743172460121184600758, −5.45476753566491114120678265571, −4.68599029193326419617218622776, −3.51404785888566711030965745770, −3.04423338321198385771390413309, −1.94900289552784900309013278949, −1.36141145079031845849197045658,
1.36141145079031845849197045658, 1.94900289552784900309013278949, 3.04423338321198385771390413309, 3.51404785888566711030965745770, 4.68599029193326419617218622776, 5.45476753566491114120678265571, 6.45910956743172460121184600758, 6.98700130116792239690493227485, 7.893113182094629940308779370867, 8.704002151935151580773315836280