L(s) = 1 | + (0.848 + 0.848i)3-s + (−0.406 + 2.61i)7-s − 1.56i·9-s − 2.56·11-s + (−2.17 − 2.17i)13-s + (−2.17 + 2.17i)17-s − 8.54·19-s + (−2.56 + 1.87i)21-s + (−5.03 + 5.03i)23-s + (3.86 − 3.86i)27-s − 5.68i·29-s + 4.79i·31-s + (−2.17 − 2.17i)33-s + (2.82 + 2.82i)37-s − 3.68i·39-s + ⋯ |
L(s) = 1 | + (0.489 + 0.489i)3-s + (−0.153 + 0.988i)7-s − 0.520i·9-s − 0.772·11-s + (−0.602 − 0.602i)13-s + (−0.526 + 0.526i)17-s − 1.95·19-s + (−0.558 + 0.408i)21-s + (−1.05 + 1.05i)23-s + (0.744 − 0.744i)27-s − 1.05i·29-s + 0.861i·31-s + (−0.378 − 0.378i)33-s + (0.464 + 0.464i)37-s − 0.590i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4114901877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4114901877\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.406 - 2.61i)T \) |
good | 3 | \( 1 + (-0.848 - 0.848i)T + 3iT^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + (2.17 + 2.17i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.17 - 2.17i)T - 17iT^{2} \) |
| 19 | \( 1 + 8.54T + 19T^{2} \) |
| 23 | \( 1 + (5.03 - 5.03i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.68iT - 29T^{2} \) |
| 31 | \( 1 - 4.79iT - 31T^{2} \) |
| 37 | \( 1 + (-2.82 - 2.82i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.54iT - 41T^{2} \) |
| 43 | \( 1 + (-0.620 + 0.620i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.848 + 0.848i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.41 + 4.41i)T - 53iT^{2} \) |
| 59 | \( 1 + 3.74T + 59T^{2} \) |
| 61 | \( 1 + 4.79iT - 61T^{2} \) |
| 67 | \( 1 + (-2.20 - 2.20i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + (3.39 + 3.39i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.80iT - 79T^{2} \) |
| 83 | \( 1 + (-5.66 - 5.66i)T + 83iT^{2} \) |
| 89 | \( 1 + 4.79T + 89T^{2} \) |
| 97 | \( 1 + (13.3 - 13.3i)T - 97iT^{2} \) |
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
−9.940215101522315535196197375343, −9.162238672177179090835023839249, −8.393447548414038732107176688713, −7.88669650115494716413049948101, −6.50916522989454263747786369771, −5.93406526951407186762414385316, −4.87126846326587141910586584277, −3.96353321192226040597996011670, −2.90343515360109230162590749112, −2.09712095013169090183560635206,
0.13893647899326867015813084062, 1.97059069266628310960631849752, 2.64828042569146568712615891199, 4.13481515930867369536530436615, 4.67746043317311753335123528602, 5.97783922365364810701172269874, 6.97788341746826635456642138935, 7.43777369301237537697566058257, 8.323655661567773789219201407732, 8.952240740557082792018383167561