| L(s) = 1 | + 1.73·3-s + 1.73i·5-s + i·11-s + 3.46i·13-s + 2.99i·15-s + 1.73i·17-s − 5.19·19-s + i·23-s + 2.00·25-s − 5.19·27-s − 4·29-s + 1.73·31-s + 1.73i·33-s − 3·37-s + 5.99i·39-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 0.774i·5-s + 0.301i·11-s + 0.960i·13-s + 0.774i·15-s + 0.420i·17-s − 1.19·19-s + 0.208i·23-s + 0.400·25-s − 1.00·27-s − 0.742·29-s + 0.311·31-s + 0.301i·33-s − 0.493·37-s + 0.960i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.530159428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530159428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 11 | \( 1 - iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 - T + 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 - 5.19iT - 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 14iT - 71T^{2} \) |
| 73 | \( 1 + 8.66iT - 73T^{2} \) |
| 79 | \( 1 - 9iT - 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 15.5iT - 89T^{2} \) |
| 97 | \( 1 - 17.3iT - 97T^{2} \) |
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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
−9.000889955171971022207157705001, −8.230651599319868757431135462737, −7.60037268864156582417049970430, −6.70755092066870205015463528233, −6.24196115642292656543714049232, −5.03752271224157640770749155658, −4.06491429003184682575964771361, −3.37703116151361071978689788192, −2.46114496763073073520910122708, −1.76337066112040734718035421638,
0.37287371486604497287843479984, 1.75972543563717985049008098268, 2.76451731965301252315267983802, 3.49017777583057608784682024149, 4.45718911169348796258652559789, 5.28129245286230549199410996406, 6.05772586572976720707087208582, 7.05257816378172898596504797683, 7.971724799551908412769546093119, 8.418427690700539258624691560443
