| L(s) = 1 | + (−1.5 + 0.866i)3-s + (1.5 − 2.59i)9-s − 3·11-s + 2·13-s + 5.19i·17-s − 5.19i·19-s − 6·23-s + 5.19i·27-s − 10.3i·29-s + 3.46i·31-s + (4.5 − 2.59i)33-s − 8·37-s + (−3 + 1.73i)39-s − 5.19i·41-s − 3.46i·43-s + ⋯ |
| L(s) = 1 | + (−0.866 + 0.499i)3-s + (0.5 − 0.866i)9-s − 0.904·11-s + 0.554·13-s + 1.26i·17-s − 1.19i·19-s − 1.25·23-s + 0.999i·27-s − 1.92i·29-s + 0.622i·31-s + (0.783 − 0.452i)33-s − 1.31·37-s + (−0.480 + 0.277i)39-s − 0.811i·41-s − 0.528i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6136873471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6136873471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + 5.19iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.735622760464094679804352445585, −8.764439734700187667739256783256, −7.960837988361317902228831190964, −6.89091377387839359008903746689, −6.04781706637437252987046953342, −5.38130014074993360084710856524, −4.37457029886652802141706842065, −3.56055383961580793173395611144, −2.06033238712800569075589944275, −0.30969000246082014815069030507,
1.29936585137234847375855601110, 2.58519152879164453411555551623, 3.93615750866165581470282315462, 5.10645051458956556813116411809, 5.68392438476744828821678133760, 6.59947152126265751554091188917, 7.48843416880746156683661506366, 8.082374327393994916182205016028, 9.160218818393113271881260638511, 10.24241117420089599362030237578
