L(s) = 1 | − 2·3-s + 7-s + 9-s − 4·11-s + 6·13-s − 2·21-s − 23-s − 5·25-s + 4·27-s − 2·29-s − 2·31-s + 8·33-s − 2·37-s − 12·39-s + 6·41-s + 4·43-s − 2·47-s + 49-s + 14·53-s + 14·59-s − 12·61-s + 63-s − 4·67-s + 2·69-s + 2·73-s + 10·75-s − 4·77-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.436·21-s − 0.208·23-s − 25-s + 0.769·27-s − 0.371·29-s − 0.359·31-s + 1.39·33-s − 0.328·37-s − 1.92·39-s + 0.937·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s + 1.82·59-s − 1.53·61-s + 0.125·63-s − 0.488·67-s + 0.240·69-s + 0.234·73-s + 1.15·75-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.272350109\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.272350109\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.52335009320829, −11.89826230916439, −11.47468043465804, −11.21899055183859, −10.73165920790014, −10.40077966368702, −10.05263327608466, −9.236712365585836, −8.901424241419214, −8.263668588457952, −7.988829023196560, −7.441834123532624, −6.870327917764560, −6.386659728620041, −5.782331277858427, −5.541031368465843, −5.324927955280179, −4.422365591118204, −4.140294227409530, −3.520839085984378, −2.862777848786254, −2.246554781798961, −1.618927524297878, −0.9368975004743434, −0.3767299082482705,
0.3767299082482705, 0.9368975004743434, 1.618927524297878, 2.246554781798961, 2.862777848786254, 3.520839085984378, 4.140294227409530, 4.422365591118204, 5.324927955280179, 5.541031368465843, 5.782331277858427, 6.386659728620041, 6.870327917764560, 7.441834123532624, 7.988829023196560, 8.263668588457952, 8.901424241419214, 9.236712365585836, 10.05263327608466, 10.40077966368702, 10.73165920790014, 11.21899055183859, 11.47468043465804, 11.89826230916439, 12.52335009320829