L(s) = 1 | − 0.639i·3-s + i·5-s + 7-s + 2.59·9-s − 4.91i·11-s + 5.17i·13-s + 0.639·15-s + 6.70·17-s + 7.89i·19-s − 0.639i·21-s − 1.23·23-s − 25-s − 3.57i·27-s − 2.97i·29-s − 3.83·31-s + ⋯ |
L(s) = 1 | − 0.369i·3-s + 0.447i·5-s + 0.377·7-s + 0.863·9-s − 1.48i·11-s + 1.43i·13-s + 0.165·15-s + 1.62·17-s + 1.81i·19-s − 0.139i·21-s − 0.257·23-s − 0.200·25-s − 0.688i·27-s − 0.553i·29-s − 0.688·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.088558082\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.088558082\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 0.639iT - 3T^{2} \) |
| 11 | \( 1 + 4.91iT - 11T^{2} \) |
| 13 | \( 1 - 5.17iT - 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 - 7.89iT - 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 + 2.97iT - 29T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 - 5.97iT - 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 + 0.0848iT - 43T^{2} \) |
| 47 | \( 1 - 9.45T + 47T^{2} \) |
| 53 | \( 1 + 4.91iT - 53T^{2} \) |
| 59 | \( 1 + 0.943iT - 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 10.4iT - 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 + 2.56iT - 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 - 4.22T + 97T^{2} \) |
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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
−9.024234925956478671190576160623, −8.062805585088519257980848766297, −7.68719011779736942824218194919, −6.67237801982516680528211086655, −6.05915052944189719451927542669, −5.22869695481458069838293694234, −3.95393914362825063737391202734, −3.46127039571591508169620633822, −2.02812693245610879665469621406, −1.15608851059673440785839185211,
0.884090581195605441633077390901, 2.06177623481522616316776761726, 3.28680987314398690187598959512, 4.26071109925170803374130092741, 5.07222969850659855966033389895, 5.48879550755112292985395983326, 6.94002150080899161739104119999, 7.43990590302148766100286300616, 8.148340729253539426668620099261, 9.188069556858212081223961597775