L(s) = 1 | + (1.5 + 2.59i)2-s + (1.5 − 2.59i)3-s + (−0.5 + 0.866i)4-s + (−1.5 − 2.59i)5-s + 9·6-s + 21·8-s + (−4.5 − 7.79i)9-s + (4.5 − 7.79i)10-s + (7.5 − 12.9i)11-s + (1.50 + 2.59i)12-s + 64·13-s − 9·15-s + (35.5 + 61.4i)16-s + (42 − 72.7i)17-s + (13.5 − 23.3i)18-s + (−8 − 13.8i)19-s + ⋯ |
L(s) = 1 | + (0.530 + 0.918i)2-s + (0.288 − 0.499i)3-s + (−0.0625 + 0.108i)4-s + (−0.134 − 0.232i)5-s + 0.612·6-s + 0.928·8-s + (−0.166 − 0.288i)9-s + (0.142 − 0.246i)10-s + (0.205 − 0.356i)11-s + (0.0360 + 0.0625i)12-s + 1.36·13-s − 0.154·15-s + (0.554 + 0.960i)16-s + (0.599 − 1.03i)17-s + (0.176 − 0.306i)18-s + (−0.0965 − 0.167i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.65384 + 0.168413i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.65384 + 0.168413i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 + 2.59i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.5 - 2.59i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (1.5 + 2.59i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-7.5 + 12.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 64T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-42 + 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (8 + 13.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-42 - 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 297T + 2.43e4T^{2} \) |
| 31 | \( 1 + (126.5 - 219. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-158 - 273. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 360T + 6.89e4T^{2} \) |
| 43 | \( 1 - 26T + 7.95e4T^{2} \) |
| 47 | \( 1 + (15 + 25.9i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (181.5 - 314. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (7.5 - 12.9i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (59 + 102. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-185 + 320. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 342T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-181 + 313. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (233.5 + 404. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 477T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-453 - 784. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 503T + 9.12e5T^{2} \) |
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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
−13.05253585052028495242296659509, −11.71474196055688402396418224015, −10.72163773940034770007008903459, −9.210009467697763846336108466812, −8.131431321827146537120156053627, −7.12422991627852628487606547170, −6.14557377251303475641347644439, −5.06508962445991743858441987979, −3.49140915880193765976326389736, −1.32128003035094388887531434919,
1.78412743762629322276598970783, 3.40932366894813089321204557860, 4.10924363295733866121086820539, 5.70546428097437895329684621541, 7.33475489140444438216219836936, 8.505395791247484752971236149756, 9.764356135350725708438517771339, 10.89723522390686879738388645702, 11.31591211882210635298441118862, 12.70896822669867857549390891060