L(s) = 1 | + 3-s + 4·5-s − 7-s + 9-s − 6·11-s − 13-s + 4·15-s + 6·17-s + 8·19-s − 21-s − 8·23-s + 11·25-s + 27-s + 2·29-s + 4·31-s − 6·33-s − 4·35-s − 2·37-s − 39-s − 4·41-s + 4·43-s + 4·45-s + 10·47-s + 49-s + 6·51-s + 6·53-s − 24·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s + 1.03·15-s + 1.45·17-s + 1.83·19-s − 0.218·21-s − 1.66·23-s + 11/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s − 1.04·33-s − 0.676·35-s − 0.328·37-s − 0.160·39-s − 0.624·41-s + 0.609·43-s + 0.596·45-s + 1.45·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 3.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.237876138\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.237876138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.260334345213482140313192853487, −7.74154154627067772258059379651, −6.98804752922337887038401100414, −5.90799267998701280173376229477, −5.53347519762181285958367893195, −4.92018020538607720513159617187, −3.50721185482607213949861297872, −2.72136776513910546304128178314, −2.18925003787028074051801537357, −1.01563783039941911278629139312,
1.01563783039941911278629139312, 2.18925003787028074051801537357, 2.72136776513910546304128178314, 3.50721185482607213949861297872, 4.92018020538607720513159617187, 5.53347519762181285958367893195, 5.90799267998701280173376229477, 6.98804752922337887038401100414, 7.74154154627067772258059379651, 8.260334345213482140313192853487