| L(s) = 1 | − 0.628·2-s − 1.60·4-s − 5-s − 4.14·7-s + 2.26·8-s + 0.628·10-s − 5.40·13-s + 2.60·14-s + 1.78·16-s + 4.14·17-s + 1.25·19-s + 1.60·20-s − 5.21·23-s + 25-s + 3.39·26-s + 6.66·28-s − 7.04·29-s + 4·31-s − 5.65·32-s − 2.60·34-s + 4.14·35-s − 7.21·37-s − 0.788·38-s − 2.26·40-s − 9.55·41-s − 6.66·43-s + 3.27·46-s + ⋯ |
| L(s) = 1 | − 0.444·2-s − 0.802·4-s − 0.447·5-s − 1.56·7-s + 0.800·8-s + 0.198·10-s − 1.49·13-s + 0.696·14-s + 0.447·16-s + 1.00·17-s + 0.288·19-s + 0.359·20-s − 1.08·23-s + 0.200·25-s + 0.665·26-s + 1.25·28-s − 1.30·29-s + 0.718·31-s − 0.999·32-s − 0.446·34-s + 0.701·35-s − 1.18·37-s − 0.127·38-s − 0.358·40-s − 1.49·41-s − 1.01·43-s + 0.482·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1630756358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1630756358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.628T + 2T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 + 7.04T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 + 9.55T + 41T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 5.21T + 59T^{2} \) |
| 61 | \( 1 + 8.29T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 - 1.25T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 19.2T + 97T^{2} \) |
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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
−8.146081759667432362001673007445, −7.51034089911066327533799451010, −6.96003042526801767751294495123, −6.01076164879897076116251135997, −5.23886262809815879089510223092, −4.49261288430424508647275262875, −3.53451702802099246623640043197, −3.09224921762201055537611079748, −1.71892685390897720260339807640, −0.22724247734669744897439356825,
0.22724247734669744897439356825, 1.71892685390897720260339807640, 3.09224921762201055537611079748, 3.53451702802099246623640043197, 4.49261288430424508647275262875, 5.23886262809815879089510223092, 6.01076164879897076116251135997, 6.96003042526801767751294495123, 7.51034089911066327533799451010, 8.146081759667432362001673007445
