| L(s) = 1 | − 1.23·3-s − 5-s − 4.23·7-s − 1.47·9-s − 2·11-s − 1.23·13-s + 1.23·15-s + 5.23·17-s − 2.23·19-s + 5.23·21-s − 7.70·23-s − 4·25-s + 5.52·27-s − 7.23·29-s + 31-s + 2.47·33-s + 4.23·35-s + 2·37-s + 1.52·39-s + 7·41-s + 3.23·43-s + 1.47·45-s − 6.47·47-s + 10.9·49-s − 6.47·51-s + 1.52·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.713·3-s − 0.447·5-s − 1.60·7-s − 0.490·9-s − 0.603·11-s − 0.342·13-s + 0.319·15-s + 1.26·17-s − 0.512·19-s + 1.14·21-s − 1.60·23-s − 0.800·25-s + 1.06·27-s − 1.34·29-s + 0.179·31-s + 0.430·33-s + 0.716·35-s + 0.328·37-s + 0.244·39-s + 1.09·41-s + 0.493·43-s + 0.219·45-s − 0.944·47-s + 1.56·49-s − 0.906·51-s + 0.209·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4065711509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4065711509\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 1.94T + 97T^{2} \) |
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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
−9.419866233878228010828378136859, −8.228806761030337811190843907246, −7.63035885361368831259174068432, −6.66466418806501796036215119455, −5.87835799111460511024950712940, −5.48220540148074145802225634706, −4.12168859895568993743511936108, −3.37460907413677230723799962819, −2.36964714832874381709068382454, −0.41054809575253654766095809891,
0.41054809575253654766095809891, 2.36964714832874381709068382454, 3.37460907413677230723799962819, 4.12168859895568993743511936108, 5.48220540148074145802225634706, 5.87835799111460511024950712940, 6.66466418806501796036215119455, 7.63035885361368831259174068432, 8.228806761030337811190843907246, 9.419866233878228010828378136859
