| L(s) = 1 | − i·2-s + i·3-s + 4-s + 6-s − 3i·8-s − 9-s + 5·11-s + i·12-s − 4i·13-s − 16-s − 4i·17-s + i·18-s + 19-s − 5i·22-s + 9i·23-s + 3·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s + 0.5·4-s + 0.408·6-s − 1.06i·8-s − 0.333·9-s + 1.50·11-s + 0.288i·12-s − 1.10i·13-s − 0.250·16-s − 0.970i·17-s + 0.235i·18-s + 0.229·19-s − 1.06i·22-s + 1.87i·23-s + 0.612·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143056195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143056195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 23 | \( 1 - 9iT - 23T^{2} \) |
| 29 | \( 1 + 7T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 5iT - 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.557460192845431505409679246555, −8.987135602512739372813957843460, −7.67397528166117069568974678175, −7.06337598321541852452035596566, −5.99023774590868564719781978101, −5.22712650114632442028959764215, −3.82581376882715718118456099489, −3.44977661335701412246732694844, −2.22330858465016496922546400472, −0.946752145016121156955186093056,
1.41547976126058926601527485918, 2.34058184010867785513644931037, 3.73041901814392379807716180843, 4.74312491393989274482022457611, 5.99950589895923952911704346261, 6.55966073514767648836525814697, 6.95403381292559703873272870824, 8.037415097275334874870375248209, 8.646729102185458122404938221479, 9.433950508057088759521013006006
