| L(s) = 1 | − 0.711·3-s + 0.887·5-s + 2.32i·7-s − 2.49·9-s − i·11-s − 0.0550i·13-s − 0.631·15-s − 0.537·17-s + (3.30 − 2.83i)19-s − 1.65i·21-s − 2.34i·23-s − 4.21·25-s + 3.91·27-s + 5.27i·29-s + 4.62·31-s + ⋯ |
| L(s) = 1 | − 0.411·3-s + 0.396·5-s + 0.880i·7-s − 0.831·9-s − 0.301i·11-s − 0.0152i·13-s − 0.163·15-s − 0.130·17-s + (0.759 − 0.650i)19-s − 0.361i·21-s − 0.488i·23-s − 0.842·25-s + 0.752·27-s + 0.980i·29-s + 0.829·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8055139539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8055139539\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + iT \) |
| 19 | \( 1 + (-3.30 + 2.83i)T \) |
| good | 3 | \( 1 + 0.711T + 3T^{2} \) |
| 5 | \( 1 - 0.887T + 5T^{2} \) |
| 7 | \( 1 - 2.32iT - 7T^{2} \) |
| 13 | \( 1 + 0.0550iT - 13T^{2} \) |
| 17 | \( 1 + 0.537T + 17T^{2} \) |
| 23 | \( 1 + 2.34iT - 23T^{2} \) |
| 29 | \( 1 - 5.27iT - 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 2.51iT - 37T^{2} \) |
| 41 | \( 1 - 12.4iT - 41T^{2} \) |
| 43 | \( 1 - 9.53iT - 43T^{2} \) |
| 47 | \( 1 + 4.22iT - 47T^{2} \) |
| 53 | \( 1 - 4.95iT - 53T^{2} \) |
| 59 | \( 1 - 6.86T + 59T^{2} \) |
| 61 | \( 1 + 7.13T + 61T^{2} \) |
| 67 | \( 1 + 3.31T + 67T^{2} \) |
| 71 | \( 1 + 8.62T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 3.87iT - 83T^{2} \) |
| 89 | \( 1 + 3.77iT - 89T^{2} \) |
| 97 | \( 1 - 17.5iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.873935081016197653929986901969, −8.306848816943641828178865677990, −7.40007389437047659459425548735, −6.37453844155020699292678480722, −5.92180798594667831036714838530, −5.23344125587406298209236849508, −4.46722051826676582120982728552, −3.07270939960702805108558097800, −2.59745869161540734431349605808, −1.25457786502486242988386464584,
0.26716270669382088069993323004, 1.54561417145654791728274998931, 2.67071618653046186915183919848, 3.71192984098622596565326327944, 4.46370170914485511807151068451, 5.57516169557550530527821461658, 5.86461645591988839820660593156, 6.92763546165945703152222519449, 7.50772227018688670958971920124, 8.333397209252195919263239833946
