L(s) = 1 | − 1.85i·3-s − 5-s + (1.29 + 2.30i)7-s − 0.432·9-s − 2.01·11-s − 1.82·13-s + 1.85i·15-s + 1.64i·17-s − 1.26i·19-s + (4.27 − 2.40i)21-s + 3.19i·23-s + 25-s − 4.75i·27-s + 4.57i·29-s − 0.834·31-s + ⋯ |
L(s) = 1 | − 1.06i·3-s − 0.447·5-s + (0.490 + 0.871i)7-s − 0.144·9-s − 0.608·11-s − 0.507·13-s + 0.478i·15-s + 0.399i·17-s − 0.289i·19-s + (0.931 − 0.525i)21-s + 0.665i·23-s + 0.200·25-s − 0.915i·27-s + 0.849i·29-s − 0.149·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.240927241\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.240927241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-1.29 - 2.30i)T \) |
good | 3 | \( 1 + 1.85iT - 3T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 1.64iT - 17T^{2} \) |
| 19 | \( 1 + 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 3.19iT - 23T^{2} \) |
| 29 | \( 1 - 4.57iT - 29T^{2} \) |
| 31 | \( 1 + 0.834T + 31T^{2} \) |
| 37 | \( 1 - 7.82iT - 37T^{2} \) |
| 41 | \( 1 - 2.75iT - 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 8.65iT - 53T^{2} \) |
| 59 | \( 1 - 3.88iT - 59T^{2} \) |
| 61 | \( 1 - 0.285T + 61T^{2} \) |
| 67 | \( 1 - 3.94T + 67T^{2} \) |
| 71 | \( 1 - 11.1iT - 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 12.8iT - 79T^{2} \) |
| 83 | \( 1 + 7.74iT - 83T^{2} \) |
| 89 | \( 1 - 8.39iT - 89T^{2} \) |
| 97 | \( 1 + 9.32iT - 97T^{2} \) |
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
−8.860539587707084057898578757527, −8.251786615484093964770504871111, −7.55548991125489581658317441473, −6.99858487848042122544540396715, −6.04952343902516773049266772562, −5.24904700115128151543362171268, −4.41064627114657480111205713936, −3.08648586031328353204439685842, −2.21303786776961700382963295917, −1.22044402760307529779135836087,
0.45838512635669763807925542563, 2.13853814797439366266323896967, 3.41372784059938647411478436529, 4.13770400379304502940189495807, 4.78340327264829670716324992688, 5.47410006219082882190411162171, 6.75801359785757385005069790183, 7.50732694806260196655609481590, 8.108636371580529260364477898570, 9.058598281616567048430999960565