L(s) = 1 | + (−1.72 + 0.131i)3-s + (2.23 − 0.0643i)5-s + (−1.19 + 0.687i)7-s + (2.96 − 0.454i)9-s + (−0.00579 − 0.0100i)11-s + (0.919 + 0.530i)13-s + (−3.85 + 0.405i)15-s + 1.57i·17-s + 6.38·19-s + (1.96 − 1.34i)21-s + (6.37 + 3.68i)23-s + (4.99 − 0.287i)25-s + (−5.06 + 1.17i)27-s + (2.66 + 4.61i)29-s + (1.15 − 2.00i)31-s + ⋯ |
L(s) = 1 | + (−0.997 + 0.0759i)3-s + (0.999 − 0.0287i)5-s + (−0.450 + 0.259i)7-s + (0.988 − 0.151i)9-s + (−0.00174 − 0.00302i)11-s + (0.254 + 0.147i)13-s + (−0.994 + 0.104i)15-s + 0.382i·17-s + 1.46·19-s + (0.429 − 0.293i)21-s + (1.32 + 0.767i)23-s + (0.998 − 0.0574i)25-s + (−0.974 + 0.226i)27-s + (0.494 + 0.856i)29-s + (0.207 − 0.359i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15133 + 0.206784i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15133 + 0.206784i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 - 0.131i)T \) |
| 5 | \( 1 + (-2.23 + 0.0643i)T \) |
good | 7 | \( 1 + (1.19 - 0.687i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.00579 + 0.0100i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.919 - 0.530i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.57iT - 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 + (-6.37 - 3.68i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.66 - 4.61i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.15 + 2.00i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 + (4.09 - 7.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.93 + 1.69i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.60 - 2.66i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 1.27iT - 53T^{2} \) |
| 59 | \( 1 + (-3.39 + 5.88i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.29 + 7.43i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.16 + 4.71i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 + 8.32iT - 73T^{2} \) |
| 79 | \( 1 + (-3.86 - 6.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.67 + 2.69i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + (0.0741 - 0.0428i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.39072498781796609740747455621, −10.63419652763719416252021291345, −9.657487100067980117753483818323, −9.104500599612459863535928442578, −7.46656410852885657350347981093, −6.48757856422428473555981910796, −5.68029060595247216558823971710, −4.86193918003165440961755151113, −3.20441283430644385273875638117, −1.38614608496536308978668374148,
1.14188732555710000839412851748, 2.98520669466063236847049372074, 4.70452366286656992101055643180, 5.57227011057042742740758841033, 6.53019862101364649007896451246, 7.25417326173495252753122747980, 8.774503504301377073876381028518, 9.897183060363345668397112161432, 10.31332736520112334278041755733, 11.41372219666366195910897872524