L(s) = 1 | − 1.53·2-s + 0.360·4-s + 2.16·5-s + 2.96·7-s + 2.51·8-s − 3.32·10-s − 11-s + 1.11·13-s − 4.55·14-s − 4.59·16-s + 6.92·17-s + 6.93·19-s + 0.780·20-s + 1.53·22-s − 0.602·23-s − 0.303·25-s − 1.71·26-s + 1.06·28-s + 4.99·29-s − 1.25·31-s + 2.01·32-s − 10.6·34-s + 6.42·35-s − 3.94·37-s − 10.6·38-s + 5.45·40-s − 1.36·41-s + ⋯ |
L(s) = 1 | − 1.08·2-s + 0.180·4-s + 0.969·5-s + 1.12·7-s + 0.890·8-s − 1.05·10-s − 0.301·11-s + 0.310·13-s − 1.21·14-s − 1.14·16-s + 1.67·17-s + 1.59·19-s + 0.174·20-s + 0.327·22-s − 0.125·23-s − 0.0606·25-s − 0.336·26-s + 0.201·28-s + 0.926·29-s − 0.224·31-s + 0.356·32-s − 1.82·34-s + 1.08·35-s − 0.648·37-s − 1.72·38-s + 0.863·40-s − 0.213·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.773471706\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.773471706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 6.93T + 19T^{2} \) |
| 23 | \( 1 + 0.602T + 23T^{2} \) |
| 29 | \( 1 - 4.99T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 + 1.36T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 - 9.97T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 67 | \( 1 + 5.56T + 67T^{2} \) |
| 71 | \( 1 - 6.01T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 3.58T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.100075218223681499787601525343, −7.62707581450196012430360743108, −6.97949027759392461658968668995, −5.73367117352204855746245170128, −5.40094389584159608331920895977, −4.61702907170420466918680099257, −3.54721021229271543130194616187, −2.44286785221980939895758803172, −1.47407885315885811244071152471, −0.960749943208063925370280448197,
0.960749943208063925370280448197, 1.47407885315885811244071152471, 2.44286785221980939895758803172, 3.54721021229271543130194616187, 4.61702907170420466918680099257, 5.40094389584159608331920895977, 5.73367117352204855746245170128, 6.97949027759392461658968668995, 7.62707581450196012430360743108, 8.100075218223681499787601525343