L(s) = 1 | − 2·4-s + 7-s − 3·11-s − 13-s + 4·16-s + 3·17-s − 4·19-s + 9·23-s − 2·28-s − 6·29-s + 2·31-s + 37-s − 3·41-s − 2·43-s + 6·44-s + 6·47-s − 6·49-s + 2·52-s − 9·53-s − 12·59-s + 5·61-s − 8·64-s + 4·67-s − 6·68-s + 9·71-s − 14·73-s + 8·76-s + ⋯ |
L(s) = 1 | − 4-s + 0.377·7-s − 0.904·11-s − 0.277·13-s + 16-s + 0.727·17-s − 0.917·19-s + 1.87·23-s − 0.377·28-s − 1.11·29-s + 0.359·31-s + 0.164·37-s − 0.468·41-s − 0.304·43-s + 0.904·44-s + 0.875·47-s − 6/7·49-s + 0.277·52-s − 1.23·53-s − 1.56·59-s + 0.640·61-s − 64-s + 0.488·67-s − 0.727·68-s + 1.06·71-s − 1.63·73-s + 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.332413061761827137499945292805, −7.82016969362823455342065727091, −6.99620031484842114987193705596, −5.89087936507254925373352317251, −5.11665245107754176961603880458, −4.65432036210041378603573414571, −3.60601386830152921539882389188, −2.71597308171493856648001012128, −1.35848013837019851368094506874, 0,
1.35848013837019851368094506874, 2.71597308171493856648001012128, 3.60601386830152921539882389188, 4.65432036210041378603573414571, 5.11665245107754176961603880458, 5.89087936507254925373352317251, 6.99620031484842114987193705596, 7.82016969362823455342065727091, 8.332413061761827137499945292805