L(s) = 1 | + 1.23·3-s + 7-s − 1.47·9-s − 4.23·11-s + 3.23·13-s + 6.47·17-s − 4.47·19-s + 1.23·21-s + 1.76·23-s − 5.52·27-s + 5·29-s + 9.70·31-s − 5.23·33-s − 3·37-s + 4.00·39-s + 9.23·41-s + 6.23·43-s − 2·47-s + 49-s + 8.00·51-s + 0.472·53-s − 5.52·57-s + 1.70·59-s + 3.70·61-s − 1.47·63-s + 0.236·67-s + 2.18·69-s + ⋯ |
L(s) = 1 | + 0.713·3-s + 0.377·7-s − 0.490·9-s − 1.27·11-s + 0.897·13-s + 1.56·17-s − 1.02·19-s + 0.269·21-s + 0.367·23-s − 1.06·27-s + 0.928·29-s + 1.74·31-s − 0.911·33-s − 0.493·37-s + 0.640·39-s + 1.44·41-s + 0.950·43-s − 0.291·47-s + 0.142·49-s + 1.12·51-s + 0.0648·53-s − 0.732·57-s + 0.222·59-s + 0.474·61-s − 0.185·63-s + 0.0288·67-s + 0.262·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.344201102\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.344201102\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 9.23T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 0.236T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 5.70T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 0.763T + 97T^{2} \) |
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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
−8.568704442062272937700776691322, −8.113525760896149142941644389840, −7.64997614619413494519724505227, −6.45374957740923811960333249577, −5.70645458785488601107201515644, −4.93974897452160256386585991469, −3.90976902771277910202252155678, −2.98081772993730869873327434177, −2.34703569094201809767295297023, −0.932723821089314837512743804676,
0.932723821089314837512743804676, 2.34703569094201809767295297023, 2.98081772993730869873327434177, 3.90976902771277910202252155678, 4.93974897452160256386585991469, 5.70645458785488601107201515644, 6.45374957740923811960333249577, 7.64997614619413494519724505227, 8.113525760896149142941644389840, 8.568704442062272937700776691322