L(s) = 1 | − 3i·3-s − 2i·7-s − 6·9-s + 5i·13-s − 6i·17-s − 6·19-s − 6·21-s − i·23-s + 9i·27-s − 9·29-s + 3·31-s − 8i·37-s + 15·39-s + 3·41-s + 8i·43-s + ⋯ |
L(s) = 1 | − 1.73i·3-s − 0.755i·7-s − 2·9-s + 1.38i·13-s − 1.45i·17-s − 1.37·19-s − 1.30·21-s − 0.208i·23-s + 1.73i·27-s − 1.67·29-s + 0.538·31-s − 1.31i·37-s + 2.40·39-s + 0.468·41-s + 1.21i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + 3iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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
−7.46878276599675555428658529202, −7.13302708999685900505024393488, −6.45117348550277483917142938497, −5.86510255429341595957904266917, −4.70519233851939142162232773833, −3.94898349350870437470422729977, −2.68444969132555361863695726691, −2.01664834644638459718230755768, −1.08831605253151223855013351164, 0,
1.97881782966950329025002164922, 3.00876975930645899658365000674, 3.71767671863564346959783769446, 4.36506737364489301577816132875, 5.29841655185222137949397838535, 5.71967753800840004977294169476, 6.43266421744379555071323253550, 7.77995806706178982265152148102, 8.468214902052962816709798002654