L(s) = 1 | + 2.18·3-s − 0.656i·5-s + i·7-s + 1.76·9-s − i·11-s + (−0.945 − 3.47i)13-s − 1.43i·15-s + 2.46·17-s − 5.98i·19-s + 2.18i·21-s − 1.56·23-s + 4.56·25-s − 2.70·27-s + 1.39·29-s − 7.01i·31-s + ⋯ |
L(s) = 1 | + 1.25·3-s − 0.293i·5-s + 0.377i·7-s + 0.586·9-s − 0.301i·11-s + (−0.262 − 0.965i)13-s − 0.369i·15-s + 0.596·17-s − 1.37i·19-s + 0.476i·21-s − 0.326·23-s + 0.913·25-s − 0.520·27-s + 0.259·29-s − 1.25i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.662375369\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.662375369\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (0.945 + 3.47i)T \) |
good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 + 0.656iT - 5T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 + 5.98iT - 19T^{2} \) |
| 23 | \( 1 + 1.56T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 7.01iT - 31T^{2} \) |
| 37 | \( 1 - 6.51iT - 37T^{2} \) |
| 41 | \( 1 + 1.43iT - 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 + 2.52iT - 47T^{2} \) |
| 53 | \( 1 + 7.93T + 53T^{2} \) |
| 59 | \( 1 + 6.65iT - 59T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 + 1.47iT - 67T^{2} \) |
| 71 | \( 1 - 4.89iT - 71T^{2} \) |
| 73 | \( 1 + 0.638iT - 73T^{2} \) |
| 79 | \( 1 + 1.86T + 79T^{2} \) |
| 83 | \( 1 + 9.14iT - 83T^{2} \) |
| 89 | \( 1 + 8.47iT - 89T^{2} \) |
| 97 | \( 1 + 9.88iT - 97T^{2} \) |
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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
−8.384248383148156660921901846011, −7.79960158858807226024789220725, −7.03090004525446559208986099682, −6.05636204177848766287053471708, −5.23295189110049131660212900474, −4.47988597599685812470734495099, −3.32026229041227983426378541684, −2.90192579958384115715345824960, −2.00426072064284423414441806018, −0.62760279284974971867946858906,
1.40301244150890045035112081413, 2.25845807826510091440332987777, 3.19530755358613899793882509580, 3.80608892128824917032743736455, 4.63251509800755293368965506465, 5.65295759696717488669359934460, 6.62947227415885120998979556445, 7.26244143643524069873522226427, 7.946949365505132129973073210412, 8.535787460579331146781263174224