L(s) = 1 | + 3-s + 2·5-s − 7-s − 2·9-s − 3·11-s + 3·13-s + 2·15-s − 2·17-s − 21-s + 3·23-s − 25-s − 5·27-s + 6·29-s − 4·31-s − 3·33-s − 2·35-s + 8·37-s + 3·39-s − 12·41-s − 7·43-s − 4·45-s − 3·47-s − 6·49-s − 2·51-s + 12·53-s − 6·55-s − 8·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.832·13-s + 0.516·15-s − 0.485·17-s − 0.218·21-s + 0.625·23-s − 1/5·25-s − 0.962·27-s + 1.11·29-s − 0.718·31-s − 0.522·33-s − 0.338·35-s + 1.31·37-s + 0.480·39-s − 1.87·41-s − 1.06·43-s − 0.596·45-s − 0.437·47-s − 6/7·49-s − 0.280·51-s + 1.64·53-s − 0.809·55-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.59244515258925376422498758981, −6.65162762267824183596954707004, −6.14483466402925691220286673793, −5.44138212952992982556681682441, −4.79272658011317298549458108042, −3.67866544312200052651527701530, −2.98574460246715112477650330635, −2.36667061146864078271392097516, −1.45266482877618482553253217746, 0,
1.45266482877618482553253217746, 2.36667061146864078271392097516, 2.98574460246715112477650330635, 3.67866544312200052651527701530, 4.79272658011317298549458108042, 5.44138212952992982556681682441, 6.14483466402925691220286673793, 6.65162762267824183596954707004, 7.59244515258925376422498758981