| L(s) = 1 | − i·3-s − 3·7-s + 2·9-s + 3i·13-s + 3·17-s + i·19-s + 3i·21-s + 9·23-s + 5·25-s − 5i·27-s − 9i·29-s − 6·31-s − 6i·37-s + 3·39-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.13·7-s + 0.666·9-s + 0.832i·13-s + 0.727·17-s + 0.229i·19-s + 0.654i·21-s + 1.87·23-s + 25-s − 0.962i·27-s − 1.67i·29-s − 1.07·31-s − 0.986i·37-s + 0.480·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.560044459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560044459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 - 9T + 23T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 3iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.452082351369450189442281696885, −9.064455935931297151545476565667, −7.77307587699736122115391753624, −7.08070584991519485848520445672, −6.50250727412608230979435762145, −5.55270838919019823822788404583, −4.35336587019880043428892678064, −3.39567998387535460812683799868, −2.24773209460051432385704753352, −0.859629915268150574693685825194,
1.09513157184347448438907192422, 3.01000527337709967171374865417, 3.46764598695459551704989318974, 4.79260310657974610597233093823, 5.43690883218433792305781071075, 6.69205073326629599870617058586, 7.16052590256910063539794378756, 8.312700185860270809450382307879, 9.338297278609692927807365341382, 9.661165550558986193511889780429
