| L(s) = 1 | − 2·4-s − 5.19·11-s − 2·13-s + 4·16-s − 5.19·17-s − 19-s − 5.19·23-s − 5·25-s − 10.3·29-s − 2·31-s + 8·37-s − 10.3·41-s − 4·43-s + 10.3·44-s + 5.19·47-s + 4·52-s + 10.3·59-s − 5·61-s − 8·64-s + 2·67-s + 10.3·68-s + 10.3·71-s + 7·73-s + 2·76-s − 4·79-s + 15.5·83-s − 10.3·89-s + ⋯ |
| L(s) = 1 | − 4-s − 1.56·11-s − 0.554·13-s + 16-s − 1.26·17-s − 0.229·19-s − 1.08·23-s − 25-s − 1.92·29-s − 0.359·31-s + 1.31·37-s − 1.62·41-s − 0.609·43-s + 1.56·44-s + 0.757·47-s + 0.554·52-s + 1.35·59-s − 0.640·61-s − 64-s + 0.244·67-s + 1.26·68-s + 1.23·71-s + 0.819·73-s + 0.229·76-s − 0.450·79-s + 1.71·83-s − 1.10·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2452394693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2452394693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−7.953304024105289850542819598740, −7.28799236158921561840126538127, −6.33798044195121852937827189449, −5.45605498291746381243154199207, −5.15301153524000825427470216468, −4.21136029630034491208465083764, −3.71759417121270089722289435276, −2.55870000690771094728438154702, −1.89305281107940542264848830881, −0.22981894718837879423769648122,
0.22981894718837879423769648122, 1.89305281107940542264848830881, 2.55870000690771094728438154702, 3.71759417121270089722289435276, 4.21136029630034491208465083764, 5.15301153524000825427470216468, 5.45605498291746381243154199207, 6.33798044195121852937827189449, 7.28799236158921561840126538127, 7.953304024105289850542819598740
