L(s) = 1 | + i·2-s − 4-s + 4.23·5-s + (−2.59 + 0.531i)7-s − i·8-s + 4.23i·10-s − i·11-s + (−0.531 − 2.59i)14-s + 16-s + 5.29·17-s − 0.250i·19-s − 4.23·20-s + 22-s + 4.50i·23-s + 12.9·25-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.89·5-s + (−0.979 + 0.200i)7-s − 0.353i·8-s + 1.33i·10-s − 0.301i·11-s + (−0.142 − 0.692i)14-s + 0.250·16-s + 1.28·17-s − 0.0573i·19-s − 0.946·20-s + 0.213·22-s + 0.938i·23-s + 2.58·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.091462082\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.091462082\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.59 - 0.531i)T \) |
| 11 | \( 1 + iT \) |
good | 5 | \( 1 - 4.23T + 5T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 0.250iT - 19T^{2} \) |
| 23 | \( 1 - 4.50iT - 23T^{2} \) |
| 29 | \( 1 - 1.05iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 2.50T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 43 | \( 1 - 7.18T + 43T^{2} \) |
| 47 | \( 1 + 7.15T + 47T^{2} \) |
| 53 | \( 1 + 4.68iT - 53T^{2} \) |
| 59 | \( 1 - 8.46T + 59T^{2} \) |
| 61 | \( 1 + 14.8iT - 61T^{2} \) |
| 67 | \( 1 - 9.42T + 67T^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 - 7.15iT - 73T^{2} \) |
| 79 | \( 1 + 0.377T + 79T^{2} \) |
| 83 | \( 1 + 1.84T + 83T^{2} \) |
| 89 | \( 1 - 1.56T + 89T^{2} \) |
| 97 | \( 1 - 7.96iT - 97T^{2} \) |
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
−9.734873860828158233947644518391, −9.080405228905874425120651189162, −8.171300495581820835794713922792, −6.97741256178110394000011464539, −6.41225480636102043578866794765, −5.56884082867383836016319027375, −5.24830369336892561715923907928, −3.59780700368762079436381445711, −2.63295065933807554501881683534, −1.25383369375988815557017479881,
1.02024913695357798336156823353, 2.25660079603352846919244387077, 2.96753851807256418847977134577, 4.20001095429789327614610504250, 5.38331094806921534736481012828, 5.99746500995490685219445763020, 6.75298409098336270136388174703, 7.916314232638488178828554965908, 9.162037754956023690500602069916, 9.497075496856794998251407303347