| L(s) = 1 | + (0.0510 + 0.000938i)2-s + (−1.90 + 1.26i)3-s + (−1.99 − 0.0733i)4-s + (−0.539 − 1.11i)5-s + (−0.0982 + 0.0628i)6-s + (0.439 − 0.178i)7-s + (−0.203 − 0.0112i)8-s + (0.852 − 2.04i)9-s + (−0.0265 − 0.0575i)10-s + (0.860 − 2.75i)11-s + (3.88 − 2.38i)12-s + (1.30 − 0.317i)13-s + (0.0225 − 0.00869i)14-s + (2.44 + 1.43i)15-s + (3.97 + 0.292i)16-s + (2.44 + 3.89i)17-s + ⋯ |
| L(s) = 1 | + (0.0361 + 0.000663i)2-s + (−1.09 + 0.731i)3-s + (−0.998 − 0.0366i)4-s + (−0.241 − 0.499i)5-s + (−0.0401 + 0.0256i)6-s + (0.166 − 0.0674i)7-s + (−0.0720 − 0.00397i)8-s + (0.284 − 0.681i)9-s + (−0.00838 − 0.0182i)10-s + (0.259 − 0.829i)11-s + (1.12 − 0.689i)12-s + (0.361 − 0.0880i)13-s + (0.00603 − 0.00232i)14-s + (0.630 + 0.371i)15-s + (0.993 + 0.0731i)16-s + (0.593 + 0.945i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.704170 - 0.0631089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.704170 - 0.0631089i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-1.52 - 4.08i)T \) |
| good | 2 | \( 1 + (-0.0510 - 0.000938i)T + (1.99 + 0.0734i)T^{2} \) |
| 3 | \( 1 + (1.90 - 1.26i)T + (1.15 - 2.76i)T^{2} \) |
| 5 | \( 1 + (0.539 + 1.11i)T + (-3.10 + 3.91i)T^{2} \) |
| 7 | \( 1 + (-0.439 + 0.178i)T + (5.01 - 4.88i)T^{2} \) |
| 11 | \( 1 + (-0.860 + 2.75i)T + (-9.03 - 6.26i)T^{2} \) |
| 13 | \( 1 + (-1.30 + 0.317i)T + (11.5 - 5.97i)T^{2} \) |
| 17 | \( 1 + (-2.44 - 3.89i)T + (-7.39 + 15.3i)T^{2} \) |
| 23 | \( 1 + (-4.47 + 2.21i)T + (13.9 - 18.2i)T^{2} \) |
| 29 | \( 1 + (5.30 + 1.71i)T + (23.5 + 16.9i)T^{2} \) |
| 31 | \( 1 + (-7.13 + 6.21i)T + (4.25 - 30.7i)T^{2} \) |
| 37 | \( 1 + (4.48 + 4.87i)T + (-3.05 + 36.8i)T^{2} \) |
| 41 | \( 1 + (2.28 + 6.44i)T + (-31.8 + 25.7i)T^{2} \) |
| 43 | \( 1 + (-6.27 - 8.87i)T + (-14.3 + 40.5i)T^{2} \) |
| 47 | \( 1 + (-5.50 - 2.84i)T + (27.1 + 38.3i)T^{2} \) |
| 53 | \( 1 + (0.475 + 10.3i)T + (-52.7 + 4.86i)T^{2} \) |
| 59 | \( 1 + (-8.95 - 1.66i)T + (55.0 + 21.1i)T^{2} \) |
| 61 | \( 1 + (-0.265 + 2.20i)T + (-59.2 - 14.4i)T^{2} \) |
| 67 | \( 1 + (1.11 - 4.75i)T + (-60.0 - 29.7i)T^{2} \) |
| 71 | \( 1 + (-0.436 - 3.63i)T + (-68.9 + 16.7i)T^{2} \) |
| 73 | \( 1 + (-5.54 + 10.4i)T + (-41.0 - 60.3i)T^{2} \) |
| 79 | \( 1 + (5.24 - 0.483i)T + (77.6 - 14.4i)T^{2} \) |
| 83 | \( 1 + (0.224 + 0.0634i)T + (70.7 + 43.4i)T^{2} \) |
| 89 | \( 1 + (0.0657 + 0.124i)T + (-50.0 + 73.6i)T^{2} \) |
| 97 | \( 1 + (-1.41 - 6.03i)T + (-86.9 + 43.0i)T^{2} \) |
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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
−11.30879513701972379417936153146, −10.52321116814579210008898430280, −9.709387979928905985524752200129, −8.672571538822776148809102969825, −7.938647848697912243294534606594, −6.09801832763550092790600133344, −5.49411110802607399764830924117, −4.47277928901504586546150953654, −3.68317766197266242427308379086, −0.77560864560760053054167314005,
1.07265565529941488235885475653, 3.26798458969449025595130698185, 4.80413814389803319591055138760, 5.46814625937592204105057113309, 6.85879334728257936250636025063, 7.33922216025536519702039190488, 8.740563887658340853140996600487, 9.627836976817338959872341046917, 10.73548908296051961014093898927, 11.62685021871169923015304560664
