| L(s) = 1 | − 1.34i·2-s + 0.192·4-s + i·7-s − 2.94i·8-s + 5.63·11-s − 4.38i·13-s + 1.34·14-s − 3.57·16-s − 2.68i·17-s − 8.38·19-s − 7.57i·22-s + 5.63i·23-s − 5.89·26-s + 0.192i·28-s + 8.32·29-s + ⋯ |
| L(s) = 1 | − 0.950i·2-s + 0.0962·4-s + 0.377i·7-s − 1.04i·8-s + 1.69·11-s − 1.21i·13-s + 0.359·14-s − 0.894·16-s − 0.652i·17-s − 1.92·19-s − 1.61i·22-s + 1.17i·23-s − 1.15·26-s + 0.0363i·28-s + 1.54·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.016540205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.016540205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 + 1.34iT - 2T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 13 | \( 1 + 4.38iT - 13T^{2} \) |
| 17 | \( 1 + 2.68iT - 17T^{2} \) |
| 19 | \( 1 + 8.38T + 19T^{2} \) |
| 23 | \( 1 - 5.63iT - 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 - 1.38iT - 43T^{2} \) |
| 47 | \( 1 + 8.58iT - 47T^{2} \) |
| 53 | \( 1 + 5.37iT - 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 1.38iT - 67T^{2} \) |
| 71 | \( 1 + 0.258T + 71T^{2} \) |
| 73 | \( 1 + 6.38iT - 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 + 16.6iT - 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 10.7iT - 97T^{2} \) |
| 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
−9.280122472752283574315645732415, −8.628043981207586890545119465214, −7.59823320621013695701320366406, −6.55528722087692496924155954711, −6.12343052090627471245994032357, −4.75962347062392047206988769003, −3.83655817688150360490591674380, −2.98637868398488236927110696357, −1.99046446173372709221805947382, −0.832905035012804275204871930223,
1.45234386963204189100433988025, 2.59905835998659698932263968360, 4.28976631084214933981951697547, 4.43359866990796846149599121740, 6.14953313583630987521159739776, 6.47218914332261267129785017919, 6.93338512994681890559312067370, 8.187850058492706224043059879585, 8.618709378667091204809157495134, 9.431166703671750755202698669850
