L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−3 + 5.19i)5-s + 7.99·8-s + (−6 − 10.3i)10-s + (6 + 10.3i)11-s − 38·13-s + (−8 + 13.8i)16-s + (63 + 109. i)17-s + (10 − 17.3i)19-s + 24·20-s − 24·22-s + (84 − 145. i)23-s + (44.5 + 77.0i)25-s + (38 − 65.8i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.268 + 0.464i)5-s + 0.353·8-s + (−0.189 − 0.328i)10-s + (0.164 + 0.284i)11-s − 0.810·13-s + (−0.125 + 0.216i)16-s + (0.898 + 1.55i)17-s + (0.120 − 0.209i)19-s + 0.268·20-s − 0.232·22-s + (0.761 − 1.31i)23-s + (0.355 + 0.616i)25-s + (0.286 − 0.496i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6605226224\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6605226224\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (3 - 5.19i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-6 - 10.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 38T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-63 - 109. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-10 + 17.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-84 + 145. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 30T + 2.43e4T^{2} \) |
| 31 | \( 1 + (44 + 76.2i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (127 - 219. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 42T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-48 + 83.1i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-99 - 171. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-330 - 571. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (269 - 465. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (442 + 765. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 792T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-109 - 188. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-260 + 450. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 492T + 5.71e5T^{2} \) |
| 89 | \( 1 + (405 - 701. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−10.27472298059006282094981156693, −9.254120021796929534445468051654, −8.453354736127816002320170989756, −7.57412715223556482736072848398, −6.92265702912014084475065900190, −6.03210777588729638093086971357, −5.03780803396016091230541940197, −4.00857799430211390603494479550, −2.78712185451989871271143488283, −1.34499351151781540492126025199,
0.21358927988003252036095267343, 1.30054712577225999602830800987, 2.69915411882516241300638816146, 3.61273600239515311305668593921, 4.82284836812762754502982698673, 5.51438297287800821489360250648, 7.05495832581985940437560931493, 7.62606402421002978639344468309, 8.638352818429540955154229683483, 9.403893733397744327525409419475