L(s) = 1 | + (−0.642 + 0.766i)3-s + (−0.342 − 0.939i)5-s + (−0.5 − 0.866i)7-s + (−0.173 − 0.984i)9-s + (0.866 + 0.5i)11-s + (−0.642 − 0.766i)13-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)17-s + (0.984 + 0.173i)21-s + (−0.939 − 0.342i)23-s + (−0.766 + 0.642i)25-s + (0.866 + 0.5i)27-s + (0.984 − 0.173i)29-s + (0.5 + 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.766i)3-s + (−0.342 − 0.939i)5-s + (−0.5 − 0.866i)7-s + (−0.173 − 0.984i)9-s + (0.866 + 0.5i)11-s + (−0.642 − 0.766i)13-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)17-s + (0.984 + 0.173i)21-s + (−0.939 − 0.342i)23-s + (−0.766 + 0.642i)25-s + (0.866 + 0.5i)27-s + (0.984 − 0.173i)29-s + (0.5 + 0.866i)31-s + (−0.939 + 0.342i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02219001756 - 0.1073014947i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02219001756 - 0.1073014947i\) |
\(L(1)\) |
\(\approx\) |
\(0.6505363779 - 0.08304430238i\) |
\(L(1)\) |
\(\approx\) |
\(0.6505363779 - 0.08304430238i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.642 + 0.766i)T \) |
| 5 | \( 1 + (-0.342 - 0.939i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.642 - 0.766i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.984 - 0.173i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.342 + 0.939i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.342 - 0.939i)T \) |
| 59 | \( 1 + (-0.984 - 0.173i)T \) |
| 61 | \( 1 + (-0.342 + 0.939i)T \) |
| 67 | \( 1 + (-0.984 + 0.173i)T \) |
| 71 | \( 1 + (-0.939 + 0.342i)T \) |
| 73 | \( 1 + (-0.766 - 0.642i)T \) |
| 79 | \( 1 + (-0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.766 + 0.642i)T \) |
| 97 | \( 1 + (0.173 - 0.984i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.52952632041631293151190665120, −24.611474861469608883575040762980, −23.80693902966187776207046688253, −22.90786221645990049864049707844, −21.92437742599996669442982334635, −21.75726652184053178933051542567, −19.65695162178898307477375649996, −19.22393081349508738329539588180, −18.51957008339767096722965194646, −17.53925213019882106988957156041, −16.61863854707220320041711636677, −15.60210941413997933518988110117, −14.51557244913336529281746732384, −13.704105788436482341454633962965, −12.31358310265444909463560669018, −11.88332423924134181037700797610, −10.9212040295253358938760872527, −9.78855541554501419964025832805, −8.496792270830708657528367520146, −7.39242708564010417774258119039, −6.37298147517188679024653374377, −5.89913081886037892608298748565, −4.22932591415554463391975621419, −2.86750956399035876035619290787, −1.74774535444846242241809682651,
0.041296146290769434272016694876, 1.07114453388113968327928686718, 3.23027043479162269924329392481, 4.3498377834736514301938287423, 4.96964425027012003566444967784, 6.294737636561172823065158758052, 7.37714090255507531286533230362, 8.695928622404065869803704275584, 9.78824055412520807948785902469, 10.30762331792897944226131518124, 11.80553499686734588496134220601, 12.222182300502677546168080697978, 13.46206600402767811011838868558, 14.62496929790807314252516988515, 15.737253390853012146568117062997, 16.398104139070821890854151196338, 17.15846286139226472255569706869, 17.872230706610720769237200291511, 19.5677030833770179523206132572, 20.15850375820730515540135042499, 20.859232760922955381951155348835, 22.09272224857885638147342332711, 22.80513939120300237610488458613, 23.48501302353989657391743444292, 24.54488449785104111229743149431