| L(s) = 1 | − 4.87·5-s + 11.7·7-s + 9.75·11-s − 22.6i·13-s − 22.1i·17-s + 17.7i·19-s − 14.1i·23-s − 1.20·25-s − 20.0·29-s + 39.7·31-s − 57.5·35-s + 2.40i·37-s − 64.3i·41-s + 3.19i·43-s + 41.8i·47-s + ⋯ |
| L(s) = 1 | − 0.975·5-s + 1.68·7-s + 0.886·11-s − 1.74i·13-s − 1.30i·17-s + 0.936i·19-s − 0.614i·23-s − 0.0480·25-s − 0.689·29-s + 1.28·31-s − 1.64·35-s + 0.0649i·37-s − 1.56i·41-s + 0.0742i·43-s + 0.890i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.791243695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.791243695\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.87T + 25T^{2} \) |
| 7 | \( 1 - 11.7T + 49T^{2} \) |
| 11 | \( 1 - 9.75T + 121T^{2} \) |
| 13 | \( 1 + 22.6iT - 169T^{2} \) |
| 17 | \( 1 + 22.1iT - 289T^{2} \) |
| 19 | \( 1 - 17.7iT - 361T^{2} \) |
| 23 | \( 1 + 14.1iT - 529T^{2} \) |
| 29 | \( 1 + 20.0T + 841T^{2} \) |
| 31 | \( 1 - 39.7T + 961T^{2} \) |
| 37 | \( 1 - 2.40iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 64.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 3.19iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 55.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 111.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 10.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 18.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 34.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 87.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 151.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 61.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 72.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 87.3T + 9.40e3T^{2} \) |
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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
−8.389334779710223137963258738376, −7.77480716682664120202868204426, −7.49677416668232937767519312999, −6.22561797281906572035863815488, −5.28188112943673680696095956506, −4.61553439325973213975812468449, −3.80221620791695766590417135700, −2.79865403663089988167507981308, −1.48699439760446070405985170807, −0.46722997535356290010529076712,
1.30120612844380640560521671066, 1.96902895931541504346726346893, 3.52243608239597483806069805019, 4.47603305726589047194542673183, 4.59153575655459085146083544182, 5.99763126925795955536159637165, 6.81775552629404105861108324645, 7.62191174786563569430971160251, 8.247125601907428056092930747898, 8.872299231646395528809061774741
