L(s) = 1 | + 0.628·2-s − 1.60·4-s − 4.14·5-s − 2.26·8-s − 2.60·10-s − 5.40·11-s + 1.78·16-s + 6.66·20-s − 3.39·22-s + 12.2·25-s + 5.65·32-s + 9.39·40-s + 1.63·41-s + 4·43-s + 8.67·44-s + 13.7·47-s − 7·49-s + 7.66·50-s + 22.4·55-s − 11.1·59-s − 7.21·61-s − 0.0277·64-s − 7.91·71-s + 14.4·79-s − 7.42·80-s + 1.02·82-s − 0.380·83-s + ⋯ |
L(s) = 1 | + 0.444·2-s − 0.802·4-s − 1.85·5-s − 0.800·8-s − 0.823·10-s − 1.62·11-s + 0.447·16-s + 1.48·20-s − 0.723·22-s + 2.44·25-s + 0.999·32-s + 1.48·40-s + 0.255·41-s + 0.609·43-s + 1.30·44-s + 1.99·47-s − 49-s + 1.08·50-s + 3.02·55-s − 1.45·59-s − 0.923·61-s − 0.00346·64-s − 0.939·71-s + 1.62·79-s − 0.829·80-s + 0.113·82-s − 0.0417·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6232158033\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6232158033\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 0.628T + 2T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 1.63T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 13.7T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 7.91T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 0.380T + 83T^{2} \) |
| 89 | \( 1 - 9.93T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.292510621843095426589830334260, −8.517340214909967714046369567915, −7.82064817105693186987922835673, −7.38915098772733660623304052117, −6.03178352607861490453876258267, −5.01122673888715230047059768815, −4.45908192042152096884951322521, −3.57862380011819886478115676700, −2.79241847977280294212214510126, −0.50365589663493428295635763476,
0.50365589663493428295635763476, 2.79241847977280294212214510126, 3.57862380011819886478115676700, 4.45908192042152096884951322521, 5.01122673888715230047059768815, 6.03178352607861490453876258267, 7.38915098772733660623304052117, 7.82064817105693186987922835673, 8.517340214909967714046369567915, 9.292510621843095426589830334260