| L(s) = 1 | + (−1.41 − i)3-s − 1.41i·5-s + (1.00 + 2.82i)9-s + 1.41·11-s − 2·13-s + (−1.41 + 2.00i)15-s − 7.07i·17-s + 4i·19-s − 7.07·23-s + 2.99·25-s + (1.41 − 5.00i)27-s − 2.82i·29-s + 6i·31-s + (−2.00 − 1.41i)33-s − 4·37-s + ⋯ |
| L(s) = 1 | + (−0.816 − 0.577i)3-s − 0.632i·5-s + (0.333 + 0.942i)9-s + 0.426·11-s − 0.554·13-s + (−0.365 + 0.516i)15-s − 1.71i·17-s + 0.917i·19-s − 1.47·23-s + 0.599·25-s + (0.272 − 0.962i)27-s − 0.525i·29-s + 1.07i·31-s + (−0.348 − 0.246i)33-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2064724351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2064724351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 + i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.07iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.356347959462174555854961889246, −7.73872740195615692871509634761, −6.88426300807374416560617874963, −6.27101526751309614562462925066, −5.18277385623651802495666436657, −4.89035482496582289109744010932, −3.69629734309122082441554295746, −2.34121596372163041027612478146, −1.30232165020607869713057838130, −0.081294069533011824491089266582,
1.60802948271258797810350980117, 2.94711158358216534346150022337, 3.95157822650082248186240223700, 4.57280219138456728305914531902, 5.65427546969965575706208879351, 6.28752490199887984473301779138, 6.91550836592658543669812400658, 7.85307765127432152674459854986, 8.806341912921317711353703486568, 9.582995781879683812113244268987
