L(s) = 1 | + (−0.642 − 0.766i)3-s + (−0.342 + 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.984 − 0.173i)17-s + (0.766 − 0.642i)19-s + (0.939 − 0.342i)21-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (0.984 − 0.173i)33-s + (−0.766 − 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)3-s + (−0.342 + 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.984 − 0.173i)17-s + (0.766 − 0.642i)19-s + (0.939 − 0.342i)21-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)27-s + (0.5 − 0.866i)29-s + 31-s + (0.984 − 0.173i)33-s + (−0.766 − 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3169189601 + 0.4715532174i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3169189601 + 0.4715532174i\) |
\(L(1)\) |
\(\approx\) |
\(0.7247128361 + 0.02787466361i\) |
\(L(1)\) |
\(\approx\) |
\(0.7247128361 + 0.02787466361i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.642 - 0.766i)T \) |
| 7 | \( 1 + (-0.342 + 0.939i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.984 - 0.173i)T \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (-0.342 - 0.939i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.984 - 0.173i)T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.50596672909024137344827637514, −19.91560709606125030679471654355, −18.79633805302039190339753029217, −18.106462854643637273569639743910, −17.318256842747119865526100673713, −16.5372221190190824026434081320, −15.952729405907269116523426808079, −15.536641273840021121353340453861, −14.16776619810120627723928314787, −13.77882519592242839980328833450, −12.79922875933233420758090142279, −11.87832813536543878693090991893, −10.99917305740359313347471041115, −10.559526747663436985479900650019, −9.85164263930692296813322162244, −8.837647779878284171734610263403, −8.128737941132194380832345078205, −6.89175632842253668979668889657, −6.23126201348223210869056620134, −5.4609235573311725412322641956, −4.41563342375582267985287903628, −3.760613603371001345070944126149, −2.98137166357359843791843168053, −1.37287292877496516991282462435, −0.25483072165418139362304447413,
1.234398153212955142990276046869, 2.249675143726030009526204997046, 2.9651075511286822051504699708, 4.44663231764328216761091850722, 5.19019261046362187325423806791, 6.159545287209185136077389202449, 6.569822240385145185944730191591, 7.70012952715506654734983583071, 8.30337636605429704383154513010, 9.36467216250117232927879772020, 10.13578890539615111546101749854, 11.240465994775360960354463917469, 11.70075814352569775577510670670, 12.52726300978692648260898678255, 13.25996260681353132801258005481, 13.73775454384871150739899677046, 15.035348914759321535650056989943, 15.80364394367339043490727037043, 16.17299608435225520052455738989, 17.5132905500332097138837268889, 17.956604809329234286159077793486, 18.376922481682139172424993058892, 19.396565145212589407548404577039, 19.917620397183488232206914494392, 20.97582245119439820154287602914