| L(s) = 1 | + 2.95i·3-s − 1.47i·5-s + (−0.700 + 2.55i)7-s − 5.71·9-s + (−2.46 + 2.21i)11-s − 13-s + 4.35·15-s + 0.856·17-s − 2.02·19-s + (−7.53 − 2.06i)21-s − 5.44·23-s + 2.82·25-s − 8.02i·27-s − 7.20i·29-s + 8.75i·31-s + ⋯ |
| L(s) = 1 | + 1.70i·3-s − 0.659i·5-s + (−0.264 + 0.964i)7-s − 1.90·9-s + (−0.743 + 0.668i)11-s − 0.277·13-s + 1.12·15-s + 0.207·17-s − 0.464·19-s + (−1.64 − 0.451i)21-s − 1.13·23-s + 0.565·25-s − 1.54i·27-s − 1.33i·29-s + 1.57i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04212294852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04212294852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.700 - 2.55i)T \) |
| 11 | \( 1 + (2.46 - 2.21i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.95iT - 3T^{2} \) |
| 5 | \( 1 + 1.47iT - 5T^{2} \) |
| 17 | \( 1 - 0.856T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 + 7.20iT - 29T^{2} \) |
| 31 | \( 1 - 8.75iT - 31T^{2} \) |
| 37 | \( 1 + 0.648T + 37T^{2} \) |
| 41 | \( 1 + 0.228T + 41T^{2} \) |
| 43 | \( 1 + 3.56iT - 43T^{2} \) |
| 47 | \( 1 + 0.0660iT - 47T^{2} \) |
| 53 | \( 1 + 8.15T + 53T^{2} \) |
| 59 | \( 1 + 8.61iT - 59T^{2} \) |
| 61 | \( 1 - 2.74T + 61T^{2} \) |
| 67 | \( 1 - 2.46T + 67T^{2} \) |
| 71 | \( 1 + 2.94T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 + 1.53iT - 89T^{2} \) |
| 97 | \( 1 - 1.74iT - 97T^{2} \) |
| 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
−8.468878172991048909641853276615, −7.978890692431668511338425207911, −6.68086025143623777274379689335, −5.76482343422670144500666406493, −5.11970341888620463905792544590, −4.68469733434112887581662752459, −3.83507291591128386571111194414, −2.93340172810235278321115494643, −2.05373537876999549040466901115, −0.01310802641914502317518850241,
1.01407504771262562983127020101, 2.13180088206133567052234112379, 2.90263302293658160239241722548, 3.73715244423717512885972161836, 4.97826172376784079936944615997, 6.05439034591306333028205215676, 6.41615109327860189133762468580, 7.20572202222559498194406176058, 7.71010423793786919755719936957, 8.176869576169498135630723640499
