L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 2·11-s − 15-s − 6·17-s − 6·19-s + 21-s − 8·23-s + 25-s + 27-s + 6·29-s − 6·31-s + 2·33-s − 35-s − 10·37-s + 2·41-s − 4·43-s − 45-s + 8·47-s + 49-s − 6·51-s − 12·53-s − 2·55-s − 6·57-s + 12·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.258·15-s − 1.45·17-s − 1.37·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.348·33-s − 0.169·35-s − 1.64·37-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s − 1.64·53-s − 0.269·55-s − 0.794·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.382497550797447782742425034866, −7.62290826463338538726072809025, −6.73228437793393096379556507042, −6.22841931485797770500089596627, −4.98024199949467462798658415046, −4.20444881363310988416251080063, −3.69084928302244391274714909733, −2.41561269657964611699072868017, −1.70120585876297520508911353739, 0,
1.70120585876297520508911353739, 2.41561269657964611699072868017, 3.69084928302244391274714909733, 4.20444881363310988416251080063, 4.98024199949467462798658415046, 6.22841931485797770500089596627, 6.73228437793393096379556507042, 7.62290826463338538726072809025, 8.382497550797447782742425034866