| L(s) = 1 | − 3.16i·3-s − i·5-s − 3.16·7-s − 7.00·9-s + 6i·13-s − 3.16·15-s − 2·17-s − 6.32i·19-s + 10.0i·21-s − 3.16·23-s − 25-s + 12.6i·27-s − 4i·29-s + 6.32·31-s + 3.16i·35-s + ⋯ |
| L(s) = 1 | − 1.82i·3-s − 0.447i·5-s − 1.19·7-s − 2.33·9-s + 1.66i·13-s − 0.816·15-s − 0.485·17-s − 1.45i·19-s + 2.18i·21-s − 0.659·23-s − 0.200·25-s + 2.43i·27-s − 0.742i·29-s + 1.13·31-s + 0.534i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.202929 + 0.489914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.202929 + 0.489914i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 3 | \( 1 + 3.16iT - 3T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 6.32iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 9.48iT - 67T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 3.16iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.807222643101876526902686763024, −8.984229153397378731215293817223, −8.260630490128092633433661250321, −7.03353269592353325942050026950, −6.69879903256589179764702686434, −5.89225463637949310957524935193, −4.41430719257763291213588108744, −2.84564328585706499638685310914, −1.81989427655183614989084688067, −0.27107672741641752521768572811,
2.96983604261534454155681560247, 3.44225408498995219433153080624, 4.53591910547356647311050971679, 5.68184582183030835364732325074, 6.28956415627370247835719815512, 7.83812829061745076779676971560, 8.720833877977333694789698027247, 9.764201852638662362074672491655, 10.18149037070771880621205414230, 10.65405309601378513942384223498
