L(s) = 1 | + (0.939 − 0.342i)3-s + (−0.939 − 1.62i)7-s + (0.766 − 0.642i)9-s + (−0.326 − 0.118i)13-s + (0.5 + 0.866i)19-s + (−1.43 − 1.20i)21-s + (−0.939 − 0.342i)25-s + (0.500 − 0.866i)27-s + (0.766 + 1.32i)31-s + 1.53·37-s − 0.347·39-s + (−0.266 + 1.50i)43-s + (−1.26 + 2.19i)49-s + (0.766 + 0.642i)57-s + (0.0603 + 0.342i)61-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)3-s + (−0.939 − 1.62i)7-s + (0.766 − 0.642i)9-s + (−0.326 − 0.118i)13-s + (0.5 + 0.866i)19-s + (−1.43 − 1.20i)21-s + (−0.939 − 0.342i)25-s + (0.500 − 0.866i)27-s + (0.766 + 1.32i)31-s + 1.53·37-s − 0.347·39-s + (−0.266 + 1.50i)43-s + (−1.26 + 2.19i)49-s + (0.766 + 0.642i)57-s + (0.0603 + 0.342i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.233455425\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.233455425\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
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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
−9.814945856854973240549199312208, −9.676298836252517872237779719920, −8.278990979610009245773470563972, −7.65541011564378512372474281495, −6.91218622448607216111525748697, −6.13799428740822469199222400872, −4.50620694711157630334599757815, −3.69656193780226036249956006023, −2.85137561760291189083279524415, −1.22796604635395947468663186980,
2.25367904216824911079160065794, 2.87578115028088518878629118637, 4.00247537330916088204788532662, 5.19957207990768060614414757973, 6.06907607959908430511468906559, 7.12962072544783067088337454329, 8.112869105366693464611177768028, 8.911415952852739180944582422477, 9.546078018103092169165542708242, 9.985069270771968773449516048955