| L(s) = 1 | + 1.17·2-s − i·3-s − 0.611·4-s + 1.67·5-s − 1.17i·6-s + (2.04 − 1.67i)7-s − 3.07·8-s − 9-s + 1.97·10-s − 3.72i·11-s + 0.611i·12-s − 0.821i·13-s + (2.40 − 1.97i)14-s − 1.67i·15-s − 2.40·16-s + 1.97·17-s + ⋯ |
| L(s) = 1 | + 0.833·2-s − 0.577i·3-s − 0.305·4-s + 0.751·5-s − 0.480i·6-s + (0.772 − 0.634i)7-s − 1.08·8-s − 0.333·9-s + 0.625·10-s − 1.12i·11-s + 0.176i·12-s − 0.227i·13-s + (0.643 − 0.528i)14-s − 0.433i·15-s − 0.600·16-s + 0.480·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77796 - 1.16585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77796 - 1.16585i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (-2.04 + 1.67i)T \) |
| 23 | \( 1 + (-4.61 - 1.31i)T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 11 | \( 1 + 3.72iT - 11T^{2} \) |
| 13 | \( 1 + 0.821iT - 13T^{2} \) |
| 17 | \( 1 - 1.97T + 17T^{2} \) |
| 19 | \( 1 - 3.72T + 19T^{2} \) |
| 29 | \( 1 + 5.79T + 29T^{2} \) |
| 31 | \( 1 - 0.566iT - 31T^{2} \) |
| 37 | \( 1 + 1.25iT - 37T^{2} \) |
| 41 | \( 1 + 5iT - 41T^{2} \) |
| 43 | \( 1 - 7.01iT - 43T^{2} \) |
| 47 | \( 1 - 5.14iT - 47T^{2} \) |
| 53 | \( 1 - 0.0649iT - 53T^{2} \) |
| 59 | \( 1 - 11.0iT - 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 - 5.03iT - 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 - 15.8iT - 73T^{2} \) |
| 79 | \( 1 - 17.6iT - 79T^{2} \) |
| 83 | \( 1 + 9.29T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
| 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
−11.11537919957446857033419786018, −9.916973817176558872090888520367, −8.979447330031220272552577178856, −8.057709876215708365510867936269, −7.04517608430075444213493731806, −5.71849109042530338184067612818, −5.40382359703099024827990891008, −3.99987901050978074414401609036, −2.88329640370077421258984030988, −1.13918705485763791823732842686,
2.04530008041290297681099442370, 3.41430851992590152471027860783, 4.68384691650905479939325912102, 5.22445353277958597992966725992, 6.06743377292351188575257231659, 7.45438877317624034315973390813, 8.715461708784207016342109391004, 9.439578054601308791683827464116, 10.07853901508939045991182634863, 11.36691701081699288894789710645
