| L(s) = 1 | − i·2-s − 4-s − 2.23i·5-s + 1.23i·7-s + i·8-s − 2.23·10-s + 4.47·11-s + i·13-s + 1.23·14-s + 16-s + 2.76i·17-s + 7.23·19-s + 2.23i·20-s − 4.47i·22-s − 7.23i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.999i·5-s + 0.467i·7-s + 0.353i·8-s − 0.707·10-s + 1.34·11-s + 0.277i·13-s + 0.330·14-s + 0.250·16-s + 0.670i·17-s + 1.66·19-s + 0.499i·20-s − 0.953i·22-s − 1.50i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684130079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684130079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 1.23iT - 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 17 | \( 1 - 2.76iT - 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 7.23iT - 23T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6.94iT - 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 6.47iT - 43T^{2} \) |
| 47 | \( 1 + 4.94iT - 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 0.472T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 8.76iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 + 9.70iT - 97T^{2} \) |
| 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
−9.607176728095004502378426959737, −8.693766088040737767552215606949, −8.469340749329121909548821295009, −7.07809524306551318955255060465, −6.07108019802061408245793658620, −5.10950365093518285800547889106, −4.29769733559596739031261590123, −3.36298921153972273060258796808, −1.96755688986959286829733024527, −0.915287690756451568720356901436,
1.24262176852662423923256635109, 3.08162627197731702120737937252, 3.77371278341399922541081479755, 4.98890474165195066997866544501, 5.92384333087476199343330538974, 6.91033823705214349534164527448, 7.21075631069730629420682867936, 8.160509047031454372191717713951, 9.262239438727417866731836792088, 9.827376562594895857013424358790
